Draft: February 26, 2010

ORDINARY PARTS OF ADMISSIBLE REPRESENTATIONS OF

p-ADIC REDUCTIVE GROUPS I. DEFINITION AND FIRST PROPERTIES

MATTHEW EMERTON

Contents

1. Introduction 1 2. Representations of p-adic analytic groups 2 3. Ordinary parts 14 4. Adjunction formulas 26 Appendix A. Functional analysis 38 References 42

1. Introduction

This paper is the first of two in which we define and study the functors of ordinary parts on categories of admissible smooth representations of p-adic reductive groups over fields of characteristic p, as well as their derived functors. These functors of ordinary parts, which are characterized as being right adjoint to parabolic induction, play an important role in the study of smooth representation theory in characteristic p. They also have important global applications: when applied in the context of the p-adically completed cohomology spaces introduced in [7], they provide a representation theoretic approach to Hida’s theory of (nearly) ordinary parts of cohomology [11, 12, 13]. The immediate applications that we have in mind are for the group GL_{2}: the results of our two papers have applications to the construction of the mod p and p-adic local Langlands correspondences for the group GL_{2}(ℚ_{p}), and to the investigation of local-global compatibility for p-adic Langlands over the group GL_{2} over ℚ.

To describe our results more specifically, let G be (the ℚ_{p}-valued points of) a connected reductive p-adic group, P a parabolic subgroup of G, \overline{P} an opposite parabolic to P, and M = P ⋂ \overline{P} the corresponding Levi factor of P and \overline{P}. If k is a finite field of characteristic p, we let Mod_{G}^{adm}(k) (resp. Mod_{M}^{adm}(k)) denote the category of admissible smooth G-representations (resp. M-representations) over k. Parabolic induction yields a functor IndG ; \overline{P} : Mod_{M}^{adm}(k) → Mod_{G}^{adm}(k). The functor of ordinary parts associated to P is then a functor Ord_{P} : Mod_{G}^{adm}(k) → Mod_{M}^{adm}(k), which is right adjoint to IndG ; \overline{P} .

In this paper, we define Ord_{P} and study its basic properties. In the sequel [9] we investigate the derived functors of Ord_{P}, and present some applications to the computation of Ext spaces in the category Mod_{G}^{adm}(k) in the case when G = GL_{2}(ℚ_{p}), computations which in turn play a role in the construction of the mod p and p-adic local Langlands correspondences (see [4] and [10]). The applications to local-global compatibility are part of the arguments of [10]. We hope to discuss the applications in the context of p-adically completed cohomology in a future paper. Let us only mention here that Theorem 3.4.8 below is an abstract formulation of Hida’s general principle that the ordinary part of cohomology should be finite over weight space.

The author was supported in part by NSF grants DMS-0401545 and DMS-0701315.

1.1. Arrangement of the paper. In order to allow for maximum flexibility in applications, we actually work throughout the paper with representations defined not just over the field k, but over general Artinian local rings, or even complete local rings, having residue field k. This necessitates a development of the foundations of the theory of admissible representations over such coefficient rings. Such a development is the subject of Section 2. We also take the opportunity in that section to present some related representation theoretic notions that do not seem to be in the literature, and which will be useful in this paper, its sequel, and future applications. The functors Ord_{P} are defined, and some of their basic properties established, in Section 3. Their characterization as adjoint functors is proved in Section 4. In the appendix we establish some simple functional analytic results about modules over p-adic integer rings.

1.2. Notation and terminology. Throughout the paper, we fix a prime p, as well as a finite extension E of ℚ_{p}, with ring of integers O. We let 𝔽 denote the residue field of O, and ϖ a choice of uniformizer of O. Let Comp(O) denote the category of complete Noetherian local O-algebras having finite residue fields, and let Art(O) denote the full subcategory of Comp(O) consisting of those objects that are Artinian (or equivalently, of finite length as O-modules).

If A is an object of Comp(O), and V is an A-module which is torsion as an O-module, then we write V^{∗} := Hom_{O}(V, E/O) to denote the Pontrjagin dual of V (where V is endowed with its discrete topology), equipped with its natural profinite topology. If V is a G-representation over A, for some group G, then the contragredient action makes V^{∗} a G-representation over A (with each element of G acting via a continuous automorphism).

If V is any O-module, then we let V_{fl} denote the maximal O-torsion free quotient of V. We write V [ϖ^{i}] to denote the kernel of multiplication by ϖ^{i} on V, and V [ϖ^{∞}] := ⋃_{i≥0} V [ϖ^{i}]. If C is any O-linear category, then for any integer i ≥ 0 (resp. i = ∞), we let C[ϖ^{i}] denote the full subcategory of C consisting of objects that are annihilated by ϖ^{i} (resp. by some power of ϖ).

1.3. Acknowledgments. I would like to thank the referee for a very careful reading of the paper, and for saving me from a blunder or two. I would also like to thank Florian Herzig for his close reading of an earlier version of the paper, and Kevin Buzzard for his helpful comments.

2. Representations of p-adic analytic groups

Let G be a p-adic analytic group. Throughout this section, we will let A denote an object of Comp(O), with maximal ideal 𝔪. We denote by Mod_{G}(A) the abelian category of representations of G over A (with morphisms being A-linear Gequivariant maps); equivalently, Mod_{G}(A) is the abelian category of A[G]-modules, where A[G] denotes the group ring of G over A. In this section we introduce, and study some basic properties of, various categories of G-representations, as well as of certain related categories of what we will call augmented G-representations.

2.1. Augmented G-representations. If H is a compact open subgroup of G, then as usual we let A[[H]] denote the completed group ring of H over A, i.e.

(2.1.1)  A[[H]] := lim_{\overleftarrow{H′}} A[H/H′],

where H′ runs over all normal open subgroups of H. We equip A[[H]] with the projective limit topology obtained by endowing each of the rings A[H/H′] appearing on the right hand side of (2.1.1) with its 𝔪-adic topology. The rings A[H/H′] are then profinite (since each H/H′ is a finite group), and hence this makes A[[H]] a profinite, and so in particular compact, topological ring.

2.1.2. Theorem. The completed group ring A[[H]] is Noetherian.

Proof. In the case when A = O = ℤ_{p}, this is proved by Lazard [14]: after replacing H by an open subgroup, if necessary, Lazard equips ℤ_{p}[[H]] with an exhaustive decreasing filtration F^{•} whose associated graded ring Gr_{F}^{•} ℤ_{p}[[H]] is isomorphic to a polynomial algebra (with generators in degree 1) over the graded ring 𝔽_{p}[t] (which is the graded ring associated to ℤ_{p} with its p-adic filtration).

For general A, we may write A[[H]] ⟶^{∼} A ˆ⊗_{ℤ_{p}}ℤ_{p}[[H]], and so equip A[[H]] with an exhaustive decreasing filtration obtained as the completed tensor product of the 𝔪-adic filtration on A and Lazard’s filtration F^{•} on ℤ_{p}[[H]]. The graded ring associated to this filtration on the completed tensor product is naturally isomorphic to a quotient of the tensor product of the individual associated graded rings, i.e. is isomorphic to a quotient of (Gr_{𝔪}^{•}A) ⊗_{𝔽_{p}[t]} Gr_{F}^{•} ℤ_{p}[[H]], where Gr_{𝔪}^{•}A denotes the graded ring associated to the 𝔪-adic filtration on A (which is also naturally an algebra over the graded ring 𝔽_{p}[t] associated to the p-adic filtration of ℤ_{p}, since p ∈ 𝔪).

Thus the associated graded ring to A[[H]] is isomorphic to a quotient of a polynomial algebra over the Noetherian ring Gr_{𝔪}^{•}A, and thus is again Noetherian. Consequently A[[H]] itself is Noetherian, as claimed. □

2.1.3. Proposition. (1) Any finitely generated A[[H]]-module admits a unique profinite topology with respect to which the A[[H]]-action on it becomes jointly continuous.

(2) Any A[[H]]-linear morphism of finitely generated A[[H]]-modules is continuous with respect to the profinite topologies on its source and target given by part (1).

Proof. Since A[[H]] is Noetherian, we may find a presentation

A[[H]]^{s} → A[[H]]^{r} → M → 0

of M, for some r, s ≥ 0. Since A[[H]] is profinite, it follows that the first arrow has closed image, and hence that M is the quotient of the profinite module A[[H]]^{r} by a closed A[[H]]-submodule. If we equip M with the induced quotient topology, then it becomes a profinite module. The resulting profinite topology on M is clearly independent of the chosen presentation, and satisfies the requirements of the proposition. □

2.1.4. Definition. If M is a finitely generated A[[H]]-module, we refer to the topology given by the preceding lemma as the canonical topology on M.

2.1.5. Definition. By an augmented representation of G over A we mean an A[G]- module M equipped with an A[[H]]-module structure for some (equivalently, any) compact open subgroup H of G, such that the two induced A[H]-actions (the first induced by the inclusion A[H] ⊂ A[[H]] and the second by the inclusion A[H] ⊂ A[G]) coincide.

2.1.6. Definition. By a profinite augmented G-representation over A we mean an augmented G-representation M over A that is also equipped with a profinite topology, so that the A[[H]]-action on M is jointly continuous ^{1} for some (equivalently any) compact open subgroup H over A.

The equivalence of the conditions “some” and “any” in the preceding definitions follows from the fact that if H_{1} and H_{2} are two compact open subgroups of G, then H := H_{1} ⋂ H_{2} has finite index in each of H_{1} and H_{2}.

We denote by Mod_{G}^{aug}(A) the abelian category of augmented G-representations over A, with morphisms being maps that are simultaneously G-equivariant and A[[H]]-linear for some (equivalently, any) compact open subgroup H of G, and by Mod_{G}^{pro aug}(A) the abelian category of profinite augmented G-representations, with morphisms being continuous A-linear G-equivariant maps (note that since A[H] is dense in A[[H]], any such map is automatically A[[H]]-linear for any compact open subgroup H of G). Forgetting the topology induces a forgetful functor

Mod_{G}^{pro aug}(A) ⟶ Mod_{G}^{aug}(A).

We let Mod_{G}^{fg aug}(A) denote the full subcategory of Mod_{G}^{aug}(A) consisting of augmented G-modules that are finitely generated over A[[H]] for some (equivalently any) compact open subgroup H of G. Proposition 2.1.3 shows that by equipping each object of Mod_{G}^{fg aug}(A) with its canonical topology, we may lift the inclusion Mod_{G}^{fg aug}(A) → Mod_{G}^{aug}(A) to an inclusion Mod_{G}^{fg aug}(A) → Mod_{G}^{pro aug}(A).

2.1.7. Proposition. The category Mod_{G}^{fg aug}(A) forms a Serre subcategory of each of the abelian categories Mod_{G}^{aug}(A) and Mod_{G}^{pro aug}(A) (i.e. it is closed under passing to subobjects, quotients, and extensions). In particular, it itself is an abelian category.

Proof. Closure under the formation of quotients and extensions is evident, and closure under the formation of subobjects follows from Theorem 2.1.2. □

2.2. Smooth G-representations. In this subsection we give the basic definitions, and state the basic results, related to smooth, and admissible smooth, representations of G over A. In the case when A is Artinian, our definitions will agree with the standard ones. Otherwise, they may be slightly unorthodox, but will be the most useful ones for our later purposes.

2.2.1. Definition. Let V be a representation of G over A. We say that a vector v ∈ V is smooth if:

(1) v is fixed by some open subgroup of G.

(2) v is annihilated by some power 𝔪^{i} of the maximal ideal of A. We let V_{sm} denote the subset of V consisting of smooth vectors.

1It is equivalent to ask that the profinite topology on M admits a neighbourhood basis at the origin consisting of A[[H]]-submodules.

2.2.2. Remark. The following equivalent way of defining smoothness can be useful: a vector v ∈ V is smooth if and only if v is annihilated by the intersection J ⋂ A[H] for some open ideal J ⊂ A[[H]], where H is some (equivalently, any) compact open subgroup of G.

2.2.3. Remark. If A is Artinian, then 𝔪^{i} = 0 for sufficiently large i, and so automatically 𝔪^{i}v = 0 for any element v of any A-module V. Thus condition (2) can be omitted from the definition of smoothness for such A, and our definition coincides with the usual one.

2.2.4. Lemma. The subset V_{sm} is an A[G]-submodule of V.

Proof. If v_{i} ∈ V_{sm} (i = 1, 2) are fixed by open subgroups H_{i} ⊂ G, and annihilated by 𝔪^{i_{1}} and 𝔪^{i_{2}}, then any A-linear combination of v_{1} and v_{2} is fixed by H_{1} ⋂ H_{2}, and annihilated by 𝔪^{max i_{1},i_{2}}, and hence also lies in V_{sm}. Thus V_{sm} is an A-submodule of V. Also, if v ∈ V_{sm} is fixed by the open subgroup H ⊂ G, and annihilated by 𝔪^{i}, then gv is fixed by gHg^{−1} ⊂ G for any g ∈ G, and again annihilated by 𝔪^{i}. Thus V_{sm} is closed under the action of G. □

2.2.5. Definition. We say that a G-representation V of G over A is smooth if V = V_{sm}; that is, if every vector of V is smooth.

We let Mod_{G}^{sm}(A) denote the full subcategory of Mod_{G}(A) consisting of smooth G-representations. The association V ↦ V_{sm} is a left-exact functor from Mod_{G}(A) to Mod_{G}^{sm}(A), which is right adjoint to the inclusion of Mod_{G}^{sm}(A) into Mod_{G}(A).

2.2.6. Lemma. The full subcategory Mod_{G}^{sm}(A) of Mod_{G}(A) is closed under the formation of subobjects, quotients, and inductive limits (and so in particular is an abelian category).

Proof. This is clear. □

The following lemma provides some useful ways of thinking about smooth representations of G.

2.2.7. Lemma. Let V be a G-representation over A, and suppose that V is torsion as an O-module. (Note that this holds automatically if A is torsion as an O-module, e.g. if A is an object of Art(O).) The following conditions on V are equivalent.

(1) V is smooth.

(2) The A-action on V and the G-action on V are both jointly continuous, when V is given its discrete topology.

(3) The A-action and the G-action on V^{∗} are both jointly continuous when V^{∗} is given its natural profinite topology.

(4) For some (equivalently, every) compact open subgroup H of G, the A[H]- action on V extends (in a necessarily unique manner) to an A[[H]]-action which is continuous when V is equipped with the discrete topology.

(5) For some (equivalently, every) compact open subgroup H of G, the H-action on V^{∗} extends (in a necessarily unique manner) to a continuous action of the completed group ring A[[H]] on V^{∗}.

(In both conditions (2) and (3), the ring A is understood to be equipped with its 𝔪-adic topology.)

Proof. The straightforward (and well-known) proofs are left to the reader. □

Lemma 2.2.7 implies that passing to Pontrjagin duals induces an anti-equivalence

(2.2.8)  \xymatrix{Mod_{G}^{sm}(A) \ar[r]^{anti ∼} & Mod_{G}^{pro aug}(A).}

2.2.9. Definition. We say that a smooth G-representation V over A is admissible if V^{H}[𝔪^{i}] (the 𝔪^{i}-torsion part of the subspace of H-fixed vectors in V) is finitely generated over A for every open subgroup H of G and every i ≥ 0.

2.2.10. Remark. If A is Artinian, so that 𝔪^{i} = 0 for i sufficiently large, then V^{H}[𝔪^{i}] = V^{H} for i sufficiently large, and so the preceding definition agrees with the usual definition of an admissible smooth A-module.

We let Mod_{G}^{adm}(A) denote the full subcategory of Mod_{G}^{sm}(A) consisting of admissible representations.

2.2.11. Lemma. A smooth G-representation V over A is admissible if and only if V^{∗} is finitely generated as an A[[H]]-module for some (equivalently, every) compact open subgroup H of G.

Proof. This is well-known. □

By Lemma 2.2.11, the anti-equivalence (2.2.8) restricts to an anti-equivalence

(2.2.12)  \xymatrix{Mod_{G}^{adm}(A) \ar[r]^{anti ∼} & Mod_{G}^{fg aug}(A).}

2.2.13. Proposition. The category Mod_{G}^{adm}(A) forms a Serre subcategory of the abelian category Mod_{G}^{sm}(A). In particular, it itself is an abelian category.

Proof. This follows directly from the anti-equivalences (2.2.8) and (2.2.12), together with Proposition 2.1.7 and Lemma 2.2.6. □

2.2.14. Remark. It follows from Lemma 2.2.11, together with the topological version of Nakayama’s lemma, that if H is an open pro-p subgroup of G (so that A[[H]] is a – typically non-commutative – local ring), then V is admissible if and only if V^{H}[𝔪] is finite dimensional over A/𝔪. In particular, if the condition of Definition 2.2.9 holds when i = 1, then it holds automatically for all values of i, and so V is admissible.

2.2.15. Definition. Let V be a G-representation over A.

(1) We say that an element v ∈ V is locally admissible if v is smooth, and if the smooth G-subrepresentation of V generated by v is admissible.

(2) We let V_{l adm} denote the subset of V consisting of locally admissible elements.

2.2.16. Lemma. The subset V_{l adm} is an A[G]-submodule of V.

Proof. Let v_{i} ∈ V_{l adm} (i = 1, 2), and let W_{i} denote the G-subrepresentation of V generated by v_{i}. Since Mod_{G}^{adm}(A) is a Serre subcategory of Mod_{G}^{sm}(A), we see that the image of the map W_{1} ⨁ W_{2} → V induced by the inclusions W_{i} ⊂ V is admissible. This image certainly contains any A-linear combination of the v_{i}, and thus V_{l adm} is an A-submodule of V. It is also clearly G-invariant, and thus is an A[G]-submodule of V. □

2.2.17. Definition. We say that a G-representation V of G over A is locally admissible if V = V_{l adm}; that is, if every vector of V is locally admissible.^{2}

We let Mod_{G}^{l adm}(A) denote the full subcategory of Mod_{G}^{sm}(A) consisting of locally admissible G-representations.

2.2.18. Proposition. The category Mod_{G}^{l adm}(A) is closed under passing to subrepresentations, quotients, and inductive limits in Mod_{G}^{sm}(A). In particular, it itself is an abelian category.

Proof. Since Mod_{G}^{adm}(A) is closed under passing to subrepresentations and quotients in Mod_{G}^{sm}(A), the same is evidently true for Mod_{G}^{l adm}(A). The closure under formation of inductive limits is also clear, since the property of being locally admissible is checked elementwise. □

2.2.19. Remark. Clearly any finitely generated locally admissible representation is in fact admissible smooth. Since any representation is the inductive limit of finitely generated ones, we see that a representation is locally admissible if and only if it can be written as an inductive limit of admissible representations.

The association V ↦ V_{l adm} gives rise to a left-exact functor from Mod_{G}(A) to Mod_{G}^{l adm}(A), which is right adjoint to the inclusion of Mod_{G}^{l adm}(A) into Mod_{G}(A).

The following diagram summarizes the relations between the various categories that we have introduced:

[figure: Mod_{G}(A); _{( )_{sm}}; _{( )_{l adm}}; Mod_{G}^{sm}(A); anti∼; Mod_{G}^{pro aug}(A); Mod_{G}^{aug}(A); Mod_{G}^{l adm}(A); Mod_{G}^{adm}(A); anti∼; Mod_{G}^{fg aug}(A); canonical topology]

2.3. Some finiteness conditions. Let Z denote the centre of G.

2.3.1. Definition. Let V be an object of Mod_{G}(A).

(1) We say that V is Z-finite if, writing I to denote the annihilator of V in A[Z], the quotient A[Z]/I is finitely generated as an A-module.

(2) We say that v ∈ V is locally Z-finite if the A[Z]-submodule of V generated by v is finitely generated as an A-module. We let V_{Z−fin} denote the subset of locally Z-finite elements of V.

(3) We say that V is locally Z-finite if every v ∈ V is locally finite over Z, i.e. if V = V_{Z−fin}.

2As was pointed out by a referee, in the context of smooth representations of p-adic groups over the field of complex numbers, such representations have been called quasi-admissible by Harish-Chandra.

2.3.2. Lemma. (1) For any object V of Mod_{G}(A), the subset V_{Z−fin} is an

A[G]-submodule of V.

(2) If V_{1} and V_{2} are two (locally) Z-finite representations, then V_{1} ⨁ V_{2} is again (locally) Z-finite.

(3) Any A[G]-invariant submodule or quotient of a (locally) Z-finite representation is again (locally) Z-finite.

Proof. If V_{1} and V_{2} are A[G]-modules annihilated by the A-finite quotients A[Z]/I_{1} and A[Z]/I_{2} of A[Z], respectively, then V_{1} ⨁ V_{2} is annihilated by A[Z]/(I_{1} ⋂ I_{2}), which is again A-finite (since it embeds into (A[Z]/I_{1})×(A[Z]/I_{2})). This proves (2) in the case of Z-finite representations.

Now suppose that v_{1} and v_{2} are two elements of V_{Z−fin}, for some A[G]-module V. Let I_{i} denote the annihilator of v_{i} in A[Z], and let W_{i} be the G-subrepresentation of V generated by v_{i}. Since Z is the centre of G, we see that I_{i} annihilates all of W_{i}. Thus each W_{i} is Z-finite, and thus so is W_{1} ⨁ W_{2}. The image of the natural map W_{1} ⨁ W_{2} → V contains all linear combinations of the v_{i}, and is G-invariant. Thus V_{Z−fin} is an A[G]-submodule of V, proving (1).

Now let V_{1} and V_{2} be locally Z-finite A[G]-modules. Then (V_{1} ⨁ V_{2})_{Z−fin} contains V_{1} and V_{2}, and so, since it is an A[G]-submodule of V_{1} ⨁ V_{2}, equals all of V_{1} ⨁ V_{2}. This proves (2) in the case of locally Z-finite representations.

Part (3) is evident (and was already used implicitly in the proof of part (1)). □

2.3.3. Lemma. If V is an object of Mod_{G}(A) which is finitely generated over A[G], then V is Z-finite if and only if V is locally Z-finite.

Proof. As we already observed in the proof of the preceding lemma, if an A[G]- module is generated by a single locally Z-finite vector, then it is Z-finite. If V is generated by finitely many such vectors, then it is a quotient of a direct sum of finitely many Z-finite representations, and so is again Z-finite (by parts (2) and (3) of the preceding lemma). □

2.3.4. Lemma. If V is an object of Mod_{G}(A), then any locally admissible vector in V is locally Z-finite.

Proof. Suppose that v ∈ V is locally admissible, let H ⊂ G be a compact open subgroup that fixes v, and let 𝔪^{i} annihilate v, for some i ≥ 0. If W ⊂ V denotes the A[G]-submodule generated by v, then W is admissible smooth, by assumption, and is annihilated by 𝔪^{i}, and so W^{H} is finitely generated over A. Since W^{H} is Z-invariant, we conclude that v is locally Z-finite, as claimed. □

2.3.5. Lemma. If an object V of Mod_{G}^{adm}(A) is finitely generated over A[G], then V is Z-finite.

Proof. This follows directly from the preceding lemmas. □

2.3.6. Lemma. Let V be an object of Mod_{G}^{sm}(A), and consider the following conditions on V.

(1) V is of finite length (as an object of Mod_{G}^{sm}(A), or equivalently, as an object of Mod_{G}(A)), and is admissible.

(2) V is finitely generated as an A[G]-module, and is admissible.

(3) V is of finite length (as an object of Mod_{G}^{sm}(A)), and is Z-finite. Then (1) implies (2) and (3).

Proof. The implication (1) ⇒ (2) is immediate. The implication (1) ⇒ (3) then follows from this together with the preceding lemma. □

We make the following conjecture:

2.3.7. Conjecture. If G is a reductive p-adic group, then the three conditions of Lemma 2.3.6 are mutually equivalent.

2.3.8. Theorem. Conjecture 2.3.7 holds in the following two cases:

(1) G is a torus.

(2) G = GL_{2}(ℚ_{p}).

Proof. Suppose first that G is a torus, so that G = Z. Let V be an object of Mod_{G}^{sm}(A). Taking into account Lemma 2.3.5, we see that each of conditions (2) and (3) of Lemma 2.3.6 implies that V is both finitely generated over A[Z] and Z-finite. These conditions, taken together, imply in their turn that V is finitely generated as an A-module. In particular, it is a finite length object of Mod_{G}^{adm}(A). Thus conditions (2) and (3) of Lemma 2.3.6 each imply condition (1).

Suppose now that G = GL_{2}(ℚ_{p}). The results of [1] and [2] imply that any irreducible smooth G-representation that is Z-finite is admissible (see [2, Cor. 1.2], and note that each of the representations listed in this corollary is admissible), and thus condition (3) of Lemma 2.3.6 implies condition (1). It remains to show that the same is true of condition (2) of Lemma 2.3.6. To this end, let V be an object of Mod_{G}^{adm}(A) that is finitely generated over A[G]. We must show that it is of finite length.

Write H = GL_{2}(ℤ_{p})Z ⊂ G. Let S ⊂ V be a finite set that generates V over A[G], and let W be the A[H]-submodule of V generated by S. Since V is smooth (by assumption), and Z-finite (by Lemma 2.3.5), we see that W is finitely generated as an A-module, annihilated by some power of 𝔪. Since W contains the generating set S, the H-equivariant embedding W ↪ V induces a G-equivariant surjection c−Ind_{H}^{G} W → V. Thus it suffices to show that any admissible quotient V of c−Ind_{H}^{G} W is of finite length as an A[G]-module. Taking into account Proposition 2.2.13, we see in fact that it suffices to prove the analogous result with W replaced by one of its (finitely many) Jordan-Hölder factors, which will be a finite dimensional irreducible representation of H over k := A/𝔪. Extending scalars to a finite extension of k, if necessary, we may furthermore assume that W is an absolutely irreducible representation of H over k, and this we do from now on.

Since V is a quotient of c−Ind_{H}^{G} W, the evaluation map

(2.3.9)  Hom_{k[G]}(c−Ind_{H}^{G} W, V) ⊗_{k} c−Ind_{H}^{G} W → V

is surjective. Lemma 2.3.10 below shows that Hom_{k[G]}(c−Ind_{H}^{G} W, V) is a finite dimensional k-vector space. It is also naturally a module over the endomorphism algebra H := End_{k[G]}(c−Ind_{H}^{G} W). In [1] it is proved that H ≅ k[T] (where T denotes a certain specified endomorphism whose precise definition need not concern us here). Thus Hom_{k[G]}(c−Ind_{H}^{G} W, V) is a finite dimensional k[T]-module, and hence is a direct sum of k[T]-modules of the form k[T]/f(T)k[T], for various nonzero polynomials f(T) ∈ k[T].

Taking into account the surjection (2.3.9), we see that it suffices to show that any G-representation of the form (c−Ind_{H}^{G} W)/f(T)(c−Ind_{H}^{G} W) is of finite length. Further extending scalars, if necessary, we may assume that f(T) factors completely in k[T] as a product of linear factors, and hence are reduced to showing that for any scalar λ ∈ k, the quotient (c−Ind_{H}^{G} W)/(T − λ)(c−Ind_{H}^{G} W) is of finite length. This, however, follows from the results of [1] and [2]. (Indeed, these quotients are either irreducible or of length two – see Theorems 1.1 and 2.7.1 of [2].) Thus the theorem is proved in the case of GL_{2}(ℚ_{p}). □

2.3.10. Lemma. If U and V are smooth representations of G over A, such that U is finitely generated over A[G] and V is admissible, then Hom_{A[G]}(U, V) is a finitely generated A-module.

Proof. Let W ⊂ U be a finitely generated A-submodule that generates U over A[G], let H ⊂ G be a compact open subgroup of G that fixes W, and let i ≥ 0 be such that 𝔪^{i} annihilates W. (Since U is a smooth representation of G over A, such H and i exist.) Restricting homomorphisms to W induces an embedding

(2.3.11)  Hom_{A[G]}(U, V) ↪ Hom_{A}(W, V^{H}[𝔪^{i}]).

Since V is an admissible smooth G-representation, its submodule V^{H}[𝔪^{i}] is finitely generated over A, and so the target of (2.3.11) is finitely generated over A. The same is thus true of the source, and the lemma follows. □

2.4. ϖ-adically admissible G-representations. In this subsection we introduce a class of representations related to the considerations of [15].

2.4.1. Definition. We say that a G-representation V over A is ϖ-adically continuous if:

(1) V is ϖ-adically separated and complete.

(2) The O-torsion subspace V [ϖ^{∞}] of V is of bounded exponent (i.e. is annihilated by a sufficiently large power of ϖ).

(3) The G-action G×V → V is continuous, when V is given its ϖ-adic topology.

(4) The A-action A×V → V is continuous, when A is given its 𝔪-adic topology and V is given its ϖ-adic topology.

2.4.2. Remark. Conditions (3) and (4) of the definition can be expressed more succinctly as follows: for any i ≥ 0, the G-action on V/ϖ^{i}V is smooth, in the sense of 2.2.1. Also, the reader may easily check, in Definition 2.4.1, that condition (3) is equivalent to the apparently weaker condition that the map G × V → V be left-continuous (i.e. that the map G → V defined by g ↦ gv is continuous for each v ∈ V) when V is equipped with its ϖ-adic topology, and that condition (4) is equivalent to the apparently weaker condition that the map A × V → V be leftcontinuous when A is equipped with its 𝔪-adic topology and V is equipped with its ϖ-adic topology.

We let Mod_{G}^{ϖ−cont}(A) denote the full subcategory of Mod_{G}(A) consisting of ϖadically continuous representations of G over A.

2.4.3. Remark. If V is a O-torsion object of Mod_{G}^{ϖ−cont}(A), then V = V [ϖ^{i}] for some i ≥ 0, hence V = V/ϖ^{i}V, and so V is in fact a smooth representation of G. Thus Mod_{G}^{ϖ−cont}(A)[ϖ^{∞}] = Mod_{G}^{sm}(A)[ϖ^{∞}]. In particular, when A is an object of Art(O), the categories Mod_{G}^{ϖ−cont}(A) and Mod_{G}^{sm}(A) coincide.

We now establish some basic results concerning the category Mod_{G}^{ϖ−cont}(A).

2.4.4. Proposition. Let 0 → V_{1} → V_{2} → V_{3} → 0 be a short exact sequence in Mod_{G}(A). If V_{2} is an object of Mod_{G}^{ϖ−cont}(A), then the following are equivalent:

(1) V_{1} is closed in the ϖ-adic topology of V_{2}, and V_{3}[ϖ^{∞}] has bounded exponent.

(2) V_{3} is an object of Mod_{G}^{ϖ−cont}(A). Furthermore, if these equivalent conditions hold, then the G-representation V_{1} is also an object of Mod_{G}^{ϖ−cont}(A), and the ϖ-adic topology on V_{2} induces the ϖ-adic topology on V_{1}.

Proof. The surjection V_{2} → V_{3} induces a surjection V_{2}/ϖ^{i}V_{2} → V_{3}/ϖ^{i}V_{3} for each i ≥ 0. Thus V_{3}/ϖ^{i}V_{3} is a smooth A[G]-representation for each i ≥ 0, since this is true of V_{2}/ϖ^{i}V_{2} by assumption. Since V_{2} is assumed to be ϖ-adically complete and separated, we see furthermore that V_{1} is closed in V_{2} if and only if V_{3} (which is isomorphic to the quotient V_{2}/V_{1}) is ϖ-adically complete and separated. From this we conclude the equivalence of (1) and (2).

If these conditions hold, then it follows from Lemma A.1 that the ϖ-adic topology on V_{2} induces the ϖ-adic topology on V_{1}, and hence that V_{1} is ϖ-adically separated and complete, since V_{2} is by assumption. This shows that V_{1} satisfies condition (1) of Definition 2.4.1. It obviously satisfies condition (2), since it is a submodule of V_{2}, which satisfies this condition by assumption. Conditions (3) and (4) are also inherited from the corresponding conditions for V_{2}, again taking into account the fact that the ϖ-adic topology on V_{2} induces the ϖ-adic topology on V_{1}. Thus V_{1} is an object of Mod_{G}^{ϖ−cont}(A), as claimed. □

2.4.5. Corollary. The category Mod_{G}^{ϖ−cont}(A) is closed under the formation of kernels and images in the abelian category Mod_{G}(A).

Proof. Suppose that 0 → V_{1} → V_{2} → V_{3} is an exact sequence of objects in Mod_{G}(A), with V_{2} and V_{3} lying in Mod_{G}^{ϖ−cont}(A). Let V′_{3} denote the image of V_{2} in V_{3}. Since V_{3}[ϖ^{∞}] has bounded exponent, by assumption, so does V′_{3}[ϖ^{∞}]. The preceding proposition then implies that both V_{1} and V′_{3} lie in Mod_{G}^{ϖ−cont}(A). This proves the closure of Mod_{G}^{ϖ−cont}(A) under the formation of kernels and images. □

2.4.6. Corollary. If V is any object of Mod_{G}^{ϖ−cont}(A), then each of V [ϖ^{∞}] and V_{fl} is also an object of Mod_{G}^{ϖ−cont}(A).

Proof. Since V [ϖ^{∞}] is of bounded exponent, the discussion of Remark A.2 show that V [ϖ^{∞}] is ϖ-adically closed in V. Since (V_{fl})[ϖ^{∞}] = 0, the result is now seen to follow directly from Proposition 2.4.4, applied to the short exact sequence 0 → V [ϖ^{∞}] → V → V_{fl} → 0. □

2.4.7. Definition. A ϖ-adically admissible representation of G over A is an object V of Mod_{G}^{ϖ−cont}(A) for which the induced G-representation on (V/ϖV)[𝔪] (which is then a smooth G-representation over A/𝔪, by Remark 2.4.2) is in fact an admissible smooth G-representation over A/𝔪.

2.4.8. Remark. Since admissible smooth representations form a Serre subcategory of Mod_{G}^{sm}(A), we see (taking into account Remarks 2.2.14 and 2.4.2) that if V is a ϖ-adically admissible G-representation over A, then in fact V/ϖ^{i}V is an admissible smooth representation for every i ≥ 0.

We let Mod_{G}^{ϖ−adm}(A) denote the full subcategory of Mod_{G}^{ϖ−cont}(A) consisting of ϖ-adically admissible representations.

2.4.9. Remark. Combining the observations of Remarks 2.4.3 and 2.4.8, one sees that Mod_{G}^{ϖ−adm}(A)[ϖ^{∞}] = Mod_{G}^{adm}(A)[ϖ^{∞}]. In particular, when A is an object of Art(O), the categories Mod_{G}^{ϖ−adm}(A) and Mod_{G}^{adm}(A) coincide.

2.4.10. Proposition. The functor V ↦ Hom_{O}(V, O) induces an A-linear antiequivalence of categories Mod_{G}^{ϖ−adm}(A)^{fl} ⟶^{∼} Mod_{G}^{fg aug}(A)^{fl}.

Proof. As is explained in the proof of [15, Thm. 1.2], the functor V ↦ Hom_{O}(V, O), the latter space being equipped with the weak topology (i.e. the topology of pointwise convergence), induces an equivalence of categories between the category of ϖ-adically separated and complete flat O-modules and the category of compact linear-topological flat O-modules. The reader may now easily check that this functor induces the required equivalence. □

2.4.11. Proposition. The category Mod_{G}^{ϖ−adm}(A) is closed under the formation of kernels, images, and cokernels in the abelian category Mod_{G}(A), and under extensions in the category Mod_{G}^{ϖ−cont}(A). In particular, Mod_{G}^{ϖ−adm}(A) is an abelian category.

Proof. Suppose first that 0 → V_{1} → V_{2} → V_{3} → 0 is a short exact sequence of objects in Mod_{G}^{ϖ−cont}(A), with V_{1} and V_{3} being objects of Mod_{G}^{ϖ−adm}(A). For any i ≥ 0, we have the exact sequence V_{1}/ϖ^{i}V_{1} → V_{2}/ϖ^{i}V_{2} → V_{3}/ϖ^{i}V_{3} → 0. Since V_{1}/ϖ^{i}V_{1} and V_{3}/ϖ^{i}V_{3} are assumed to be admissible smooth G-representations over A, the same is true of V_{2}/ϖ^{i}V_{2}, by Proposition 2.2.13. Thus V_{2} is an object of Mod_{G}^{ϖ−adm}(A).

Suppose now that 0 → V_{1} → V_{2} → V_{3} is an exact sequence of objects of Mod_{G}(A), with V_{2} and V_{3} being objects of Mod_{G}^{ϖ−adm}(A). Let V′_{3} denote the image of V_{2} in V_{3}. It follows from Corollary 2.4.5 that V_{1} and V′_{3} are objects of Mod_{G}^{ϖ−cont}(A). Since V_{3}[ϖ^{∞}] is an object of Mod_{G}^{ϖ−adm}(A)[ϖ^{∞}], which by Remark 2.4.9 equals Mod_{G}^{adm}(A)[ϖ^{∞}], so is V′_{3}[ϖ^{∞}], by Proposition 2.2.13. Choose i ≥ 0 so that V′_{3}[ϖ^{i}] = V′_{3}[ϖ^{∞}]. For each j ≥ i, our given exact sequence yields an exact sequence

V′_{3}[ϖ^{i}] = V′_{3}[ϖ^{j}] → V_{1}/ϖ^{j}V_{1} → V_{2}/ϖ^{j}V_{2} → V′_{3}/ϖ^{j}V′_{3} → 0.

Since the first and third terms of the exact sequence lie in Mod_{G}^{adm}(A), we conclude from Proposition 2.2.13 that the other two terms do also. We conclude that V_{1} and V′_{3} are both objects of Mod_{G}^{ϖ−adm}.

Suppose finally that V_{1} → V_{2} → V_{3} → 0 is an exact sequence of objects of Mod_{G}(A), with V_{1} and V_{2} being objects of Mod_{G}^{ϖ−adm}(A). We wish to show that V_{3} is also an object of Mod_{G}^{ϖ−adm}(A). By what we have already proved, we know that the image of V_{1} in V_{2} again lies in Mod_{G}^{ϖ−adm}(A). Thus, replacing V_{1} by this image, we may assume that our sequence is in fact short exact.

We intend to apply Theorem A.11, but before doing so, we must verify that V_{3} is complete, with O-torsion of bounded exponent. Since V_{2}[ϖ^{∞]} has bounded exponent by assumption, it is no loss of generality to do this after replacing V_{1} and V_{2} by (V_{1})_{fl} and (V_{2})_{fl} respectively, and replacing V_{3} by its quotient by the image of V_{2}[ϖ^{∞}]. We may thus assume that V_{1} and V_{2} are O-torsion free.

Applying Hom_{O}(– , O) to the embedding V_{1} ↪ V_{2}, and taking into account the anti-equivalence of categories of Proposition 2.4.10, we obtain a map M_{2} → M_{1} of objects of Mod_{G}^{fg aug}(A)^{fl}. Let M denote the image of M_{2} in M_{1}, and let M̃ denote the saturation of M in M_{1} as a O-module, i.e.

M̃ = {m ∈ M_{1} | ϖ^{i}m ∈ M for some i ≥ 0}.

The quotient M_{1}/M̃ is again an object of Mod_{G}^{fg aug}(A)^{fl}, and thus the complex M_{2} → M_{1} → M_{1}/M̃ in Mod_{G}^{fg aug}(A)^{fl}, with the last map being a surjection, arises via the anti-equivalence of Proposition 2.4.10 from a complex V → V_{1} → V_{2} in Mod_{G}^{ϖ−adm}(A)^{fl}, with the first arrow being an injection, by Lemma A.13. Since the second arrow is also an injection, by assumption, we conclude that V = 0, and hence that M_{1}/M̃ = 0, i.e. that the saturation in M_{1} of the image M of M_{2} is equal to M_{1}. Thus the quotient M_{1}/M is O-torsion. Since it is furthermore finitely generated over A[[H]], for any compact open subgroup H of G (as this is true of M_{1}), it is of bounded exponent. It follows from Lemma A.14 that V_{3}[ϖ^{∞}] has bounded exponent. Lemma A.1 then implies that ϖ-adic topology on V_{2} induces the ϖ-adic topology on V_{1}. Since V_{1} is ϖ-adically complete by assumption, we find that it is ϖ-adically closed in V_{2}, and so it follows from Proposition 2.4.4 that V_{3} is an object of Mod_{G}^{ϖ−cont}(A), and in particular is ϖ-adically complete.

We now apply Hom_{O}(– , O) to the exact sequence 0 → V_{1} → V_{2} → V_{3} → 0 to obtain (by Theorem A.11) the exact sequence

0 → Hom_{O}((V_{3})_{fl}, O) → Hom_{O}((V_{2})_{fl}, O) → Hom_{O}((V_{1})_{fl}, O)

→ Hom_{O}(V_{3}[ϖ^{∞}], E/O) → Hom_{O}(V_{2}[ϖ^{∞}], E/O).

(Note that Hom_{O}((V_{i})_{fl}, O) ⟶^{∼} Hom_{O}(V_{i}, O) for i = 1, 2, 3, since O is torsion free as a module over itself.) Proposition 2.4.10 and Lemma 2.2.11 (together with Remark 2.4.9) imply that, for any compact open subgroup H of G, each of Hom_{O}((V_{1})_{fl}, O), Hom_{O}((V_{2})_{fl}, O), and Hom_{O}(V_{2}[ϖ^{∞}], E/O) is a finitely generated A[[H]]-module. Since A[[H]] is Noetherian, a consideration of this exact sequence shows that the A[[H]]-modules Hom_{O}((V_{3})_{fl}, O) and Hom_{O}(V_{3}[ϖ^{∞}], E/O) are also finitely generated, and thus that (V_{3})_{fl} and V_{3}[ϖ^{∞}] are both objects of Mod_{G}^{ϖ−adm}(A) (by the same proposition and lemma). We have already observed that since V_{3}[ϖ^{∞}] has bounded exponent, Proposition 2.4.4 implies that V_{3} lies in Mod_{G}^{ϖ−cont}(A). Since it is furthermore an extension of the objects (V_{3})_{fl} and V_{3}[ϖ^{∞}] of Mod_{G}^{ϖ−adm}(A), it follows from the first paragraph of the proof that V_{3} is itself an object of Mod_{G}^{ϖ−adm}(A). □

2.4.12. Proposition. If 0 → V_{1} → V_{2} → V_{3} → 0 is a short exact sequence in Mod_{G}(A), and if V_{2} is an object of Mod_{G}^{ϖ−adm}(A), then the following are equivalent:

(1) V_{1} is closed in the ϖ-adic topology of V_{2}, and V_{3}[ϖ^{∞}] has bounded exponent.

(2) V_{1} is an object of Mod_{G}^{ϖ−adm}(A).

(3) V_{3} is an object of Mod_{G}^{ϖ−cont}(A).

(4) V_{3} is an object of Mod_{G}^{ϖ−adm}(A). If these equivalent conditions hold, then the ϖ-adic topology on V_{2} induces the ϖadic topology on V_{1}.

Proof. The equivalence of (1) and (3) follows from Proposition 2.4.4, and clearly (4) implies (3). Since V_{1} (resp. V_{3}) is the kernel of the map V_{2} → V_{3} (resp. the cokernel of the map V_{1} → V_{2}), the equivalence of (2) and (4) follows from Proposition 2.4.11. Finally, note that if (3) holds, then V_{3}/ϖ^{i}V_{3} is a quotient of V_{2}/ϖ^{i}V_{2} for any i ≥ 0. The latter module is an admissible A[G]-representation by assumption, and thus so is the former. Consequently, V_{3} is an object of Mod_{G}^{ϖ−adm}(A), and so (3) implies (4). The remaining claim of the proposition follows from Proposition 2.4.4. □

2.4.13. Corollary. If V is any object of Mod_{G}^{ϖ−adm}(A), then each of V [ϖ^{∞}] and V_{fl} is also an object of Mod_{G}^{ϖ−adm}(A).

Proof. This follows directly from the preceding proposition, applied to the short exact sequence 0 → V [ϖ^{∞}] → V → V_{fl} → 0. (Compare the proof of Corollary 2.4.6.)

□

3. Ordinary parts

In this section we take G to be a p-adic reductive group (by which we always mean the group of ℚ_{p}-points of a connected reductive algebraic group over ℚ_{p}). We suppose that P is a parabolic subgroup of G (more precisely, the group of ℚ_{p}-valued points of a parabolic subgroup of G defined over ℚ_{p}), with unipotent radical N, that M is a Levi factor of P (so that P = MN), and that \overline{P} is an opposite parabolic to P, with unipotent radical \overline{N}, chosen so that M = P ⋂ \overline{P}. (This condition uniquely determines \overline{P}.) Our goal in this section is to define the functor of ordinary parts, which (for any object A of Comp(O)) is a functor Ord_{P} : Mod_{G}^{adm}(A) → Mod_{M}^{adm}(A). In the following section we will show that it is right adjoint to parabolic induction IndG ; \overline{P} : Mod_{M}^{adm}(A) → Mod_{G}^{adm}(A).

In fact, it is technically convenient to define Ord_{P} in a more general context, as a functor Ord_{P} : Mod_{P}^{sm}(A) → Mod_{M}^{sm}(A), and this we do in Subsection 3.1. In Subsection 3.2 we establish some elementary properties of Ord_{P}. In Subsection 3.3, we prove that Ord_{P}, when applied to admissible smooth G-representations, yields admissible smooth M-representations. Finally, Subsection 3.4 extends the main definitions and results of the preceding subsections to the context of ϖ-adically continuous and ϖ-adically admissible representations over A.

The functor Ord_{P} is analogous to the locally analytic Jacquet functor J_{P} studied in [6] and [8], and many of the constructions and arguments of this section and the next are suitable modifications of the constructions and arguments found in those papers. Typically, the arguments become simpler. (As a general rule, the theory of Ord_{P} is more elementary than the theory of the Jacquet functors J_{P}, just as the theory of p-adic modular forms is more elementary in the ordinary case than in the more general finite slope case.)

3.1. The definition of Ord_{P}. We fix G, P, \overline{P}, etc., as in the preceding discussion. We let A denote an object of Comp(O).

3.1.1. Definition. If V is a representation of N over A, and if N_{1} ⊂ N_{2} are compact open subgroups of N, then let h_{N_{2},N_{1}} : V^{N_{1}} → V^{N_{2}} denote the operator defined by h_{N_{2},N_{1}}(v) := ∑_{n∈N_{2}/N_{1}} nv. (Here we are using n to denote both an element of N_{2}/N_{1}, and a choice of coset representative for this element in N_{2}. Since v ∈ V^{N_{1}}, the value nv is well-defined independently of the choice of coset representative.)

3.1.2. Lemma. If N_{1} ⊂ N_{2} ⊂ N_{3} are compact open subgroups of N, then

h_{N_{3},N_{2}}h_{N_{2},N_{1}} = h_{N_{3},N_{1}}.

Proof. This is immediate from the definition of the operators. □

Fix a compact open subgroup P_{0} of P, and set M_{0} := M ⋂ P_{0}, N_{0} := N ⋂ P_{0}, and M^{+} := {m ∈ M | mN_{0}m^{−1} ⊂ N_{0}}. Let Z_{M} denote the centre of M, and write Z_{M}^{+} := M^{+} ⋂ Z_{M}. Note that each of M_{0} and N_{0} is a subgroup of G, while M^{+} and Z_{M}^{+} are submonoids of G.

3.1.3. Definition. If V is a representation of P over A, then for any m ∈ M^{+}, we define h_{N_{0},m} : V^{N_{0}} → V^{N_{0}} via the formula h_{N_{0},m}(v) := h_{N_{0},mN_{0}m^{−1}}(mv).

3.1.4. Lemma. If m_{1}, m_{2} ∈ M^{+}, then h_{N_{0},m_{1}m_{2}} = h_{N_{0},m_{1}}h_{N_{0},m_{2}}.

Proof. We compute that

h_{N_{0},m_{1}m_{2}} = h_{N_{0},m_{1}m_{2}N_{0}m_{2}^{−1}m_{1}^{−1}}m_{1}m_{2}

= h_{N_{0},m_{1}N_{0}m_{1}^{−1}}h_{m_{1}N_{0}m_{1}^{−1},m_{1}m_{2}N_{0}m_{2}^{−1}m_{1}^{−1}}m_{1}m_{2}

= h_{N_{0},m_{1}N_{0}m_{1}^{−1}}m_{1}h_{N_{0},m_{2}N_{0}m_{2}^{−1}}m_{2} = h_{N_{0},m_{1}}h_{N_{0},m_{2}}.

□

If m ∈ M_{0}, then (since in this case mN_{0}m^{−1} = N_{0}) the operator h_{N_{0},m} coincides with the given action of m on V^{N_{0}}. In general, the preceding lemma shows that the operators h_{N_{0},m} induce an action of the monoid M^{+} on V^{N_{0}}, which we refer to as the Hecke action of M^{+}.

Regarding V^{N_{0}} as a module over the monoid algebra A[M^{+}], and so in particular its central subalgebra A[Z_{M}^{+}], via this action, we may consider the A[Z_{M}]-module Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}), as well as its submodule Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}})_{Z_{M} −fin}.

3.1.5. Lemma. If U is an A[Z_{M}^{+}]-module which is finitely generated over A, then evaluation at the element 1 ∈ Z_{M} induces an embedding Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], U) ↪ U. In particular, Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], U) is finitely generated over A.

Proof. If B denotes the image of A[Z_{M}^{+}] in End_{A}(U), then B is a commutative finite A-algebra, and so is isomorphic to the product of its localizations at its finitely many maximal ideals, i.e. B ⟶^{∼} ∏_{𝔪 maximal} B_{𝔪}. This factorization of B induces a corresponding factorization of the B-module U, namely U = ∏_{𝔪 maximal} U_{𝔪}, and a consequent factorization

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], U) ⟶^{∼} ∏_{𝔪 maximal} Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], U_{𝔪}).

We will say that a maximal ideal 𝔪 is ordinary (resp. non-ordinary) if and only if no (resp. some) element z ∈ Z_{M}^{+} has its image in B lying in 𝔪. If 𝔪 is ordinary, then the A[Z_{M}^{+}]-module structure on U_{𝔪} extends in a unique manner to an A[Z_{M}]- module structure (since Z_{M}^{+} generates Z_{M} as a group [6, Prop. 3.3.2 (i)]), and evaluation at 1 ∈ Z_{M} induces an isomorphism

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], U_{𝔪}) ⟶^{∼} U_{𝔪}.

On the other hand, if 𝔪 is non-ordinary, then it is easily seen that

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], U_{𝔪}) = 0.

Thus evaluation at 1 induces an embedding

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], U) ⟶^{∼} ∏_{𝔪 ordinary} U_{𝔪} ⊂ U,

as claimed. □

3.1.6. Lemma. Let W be an A[Z_{M}^{+}]-module, and let ϕ ∈ Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W).

(1) The image im(ϕ) of ϕ is an A[Z_{M}^{+}]-submodule of W.

(2) ϕ is locally Z_{M}-finite if and only if im(ϕ) is finitely generated as an Amodule.

(3) If z ∈ Z_{M}^{+}, then z im(ϕ) = im(ϕ).

Proof. Claim (1) is clear, since we have z_{1}ϕ(z_{2}) = ϕ(z_{1}z_{2}) for z_{1} ∈ Z_{M}^{+} and z_{2} ∈ Z_{M}. As for claim (2), note that we may regard ϕ as an element of the A[Z_{M}]-submodule Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], im(ϕ)) of Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W). If im(ϕ) is finitely generated over A, then Lemma 3.1.5 shows that this submodule is finitely generated over A, and thus that ϕ is locally Z_{M}-finite. On the other hand, let ev_{1} : Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W)_{Z_{M} −fin} → W denote the map given by evaluation at 1 ∈ Z_{M}^{+}. For any ϕ ∈ Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W)_{Z_{M} −fin}, if U denotes the A[Z_{M}]- submodule of Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W)_{Z_{M} −fin} generated by ϕ, then clearly ev_{1}(U) coincides with the image im(ϕ) of ϕ. Thus if ϕ is locally Z_{M}-finite, we see that U is finitely generated over A, and hence the same is true of im(ϕ). This completes the proof of (2). Claim (3) follows from the following calculation:

im(ϕ) := A-linear span of the elements ϕ(z′), z′ ∈ Z_{M}
= A-linear span of the elements ϕ(zz′), z′ ∈ Z_{M}
= A-linear span of the elements zϕ(z′), z′ ∈ Z_{M}
= z im(ϕ).
□

3.1.7. Lemma. Let W be an A[M^{+}]-module.

(1) The A[Z_{M}]-module structure on Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W) extends naturally to an A[M]-module structure, characterized by the property that for any z ∈ Z_{M}, the map Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W) → W induced by evaluation at z is M^{+}-equivariant.

(2) If the M_{0}-action on W is furthermore smooth, then the induced M-action on Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W)_{Z_{M} −fin} is smooth.

Proof. From [6, Prop. 3.3.6], we see that the natural map A[M^{+}] ⊗_{A[Z_{M}^{+} ]} A[Z_{M}] → A[M] is an isomorphism, and thus that the restriction map induces an isomorphism

(3.1.8)  Hom_{A[M+]}(A[M], W) ⟶^{∼} Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W).

The source of this isomorphism has a natural A[M]-module structure, which we may then transport to the target. This A[M]-module structure clearly satisfies the condition of (1).

Suppose now that W is M_{0}-smooth. Part (4) of Lemma 2.2.7 shows that the A[M_{0}]-action on W then extends to an A[[M_{0}]]-action, and that any element of W is annihilated by an open ideal in A[[M_{0}]]. Let ϕ be an element of Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W)_{Z_{M} −fin}. Since im(ϕ) is a finitely generated A-submodule of W, by Lemma 3.1.6, we may find an open ideal I in A[[M_{0}]] which annihilates im(ϕ). Hence the intersection I ⋂ A[M_{0}] annihilates ϕ itself (as follows directly from the definition of the M-action on ϕ, together with the fact that M_{0} ⊂ M^{+}), and so the M_{0}-action on ϕ is smooth (by Remark 2.2.2). Thus the M-action on Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W)_{Z_{M} −fin} is smooth, proving (2). □

We now give the main definition of this section.

3.1.9. Definition. If V is a smooth representation of P over A, then we write

Ord_{P} (V) := Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}})_{Z_{M} −fin},

and refer to Ord_{P} (V) as the P -ordinary part of V (or just the ordinary part of V, if P is understood).

Lemma 3.1.7 shows that Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}), and hence Ord_{P} (V), is naturally an M-representation over A. Since the Hecke M_{0}-action on V^{N_{0}} coincides with the given M_{0}-action, it is smooth, and so the same lemma shows that the M-action on Ord_{P} (V) is smooth. Thus the formation of Ord_{P} (V) yields a functor from Mod_{P}^{sm}(A) to Mod_{M}^{sm}(A).

3.1.10. Definition. We define the canonical lifting

Ord_{P} (V) → V^{N_{0}}

to be the composite of the inclusion Ord_{P} (V) ⊂ Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}) with the map Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}) → V^{N_{0}} given by evaluation at the element 1 ∈ A[Z_{M}].

Our choice of terminology is motivated by that of [3, §4].

Although the definition of the functor Ord_{P} depends on the choice of compact open subgroup P_{0} of P, we now show that it is independent of this choice, up to natural isomorphism. To this end, suppose that P′_{0} is an open subgroup of P_{0}. Write M′_{0} := M ⋂ P′_{0}, N′_{0} := N ⋂ P′_{0}, (M^{+})′ := {m ∈ M | mN′_{0}m^{−1} ⊂ N′_{0}}, (Z_{M}^{+})′ := (M^{+})′ ⋂ Z_{M}. We define the Hecke action of (M^{+})′ on V^{N′_{0}} in analogy to the Hecke action of M^{+} on V^{N_{0}}.

Let z ∈ Z_{M} be such that zN′_{0}z^{−1} ⊂ N_{0}, and for any object V of Mod_{P}^{sm}(A), define a map h_{N_{0},N′_{0},z} : V^{N′_{0}} → V^{N_{0}} via the formula v ↦ h_{N_{0},zN′_{0}z^{−1}}zv, for v ∈ V^{N′_{0}}.

3.1.11. Lemma. (1) The map h_{N_{0},N′_{0},z} intertwines the Hecke action of the in-

tersection M^{+} ⋂(M^{+})′ on its source and target.

(2) If P′′_{0} is another compact open subgroup of P, giving rise to N′′_{0}, (M^{+})′′, etc., and if z, z′ ∈ Z_{M} are such that zN′_{0}z^{−1} ⊂ N_{0} and z′N′′_{0}(z′)^{−1} ⊂ N′_{0}, then h_{N_{0},N′_{0},z}h_{N′_{0},N′′_{0},z′} = h_{N_{0},N′′_{0},zz′}.

Proof. If m ∈ M^{+} ⋂(M^{+})′, then

h_{N_{0},m}h_{N_{0},N′_{0},z} = h_{N_{0},mN_{0}m^{−1}}mh_{N_{0},zN′_{0}z^{−1}}z
= mh_{m^{−1}N_{0}m,N_{0}}h_{N_{0},zN′_{0}z^{−1}}z
= mh_{m^{−1}N_{0}m,zN′_{0}z^{−1}}z
= zmh_{(zm)^{−1}N_{0}zm,N′_{0}}
= zmh_{(zm)^{−1}N_{0}zm,m^{−1}N′_{0}m}h_{m^{−1}N′_{0}m,N′_{0}}
= zh_{z^{−1}N_{0}z,N′_{0}}h_{N′_{0},mN′_{0}m^{−1}}m
= h_{N_{0},zN′_{0}z^{−1}}zh_{N′_{0},mN′_{0}m^{−1}}m
= h_{N_{0},N′_{0},z}h_{N′_{0},m}.

This proves claim (1). Claim (2) is proved by an analogous computation. □

If we write Y′ = M^{+} ⋂(M^{+})′ and Y = Z_{M}^{+} ⋂(Z_{M}^{+})′, then Y generates Z_{M} as a group [6, Prop. 3.3.2 (i)], and Y′ and Z_{M} generate M [6, Prop. 3.3.2 (ii)]. In particular, the inclusions

Hom_{A[(Z_{M}^{+} )′]}(A[Z_{M}], V^{N′_{0}}) ⊂ Hom_{A[Y]}(A[Z_{M}], V^{N′_{0}})

and

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}) ⊂ Hom_{A[Y]}(A[Z_{M}], V^{N_{0}})

are actually equalities. Since h_{N_{0},N′_{0},z} is Y′-equivariant, it induces an M = Y′Z_{M}- equivariant map

Hom_{A[Y]}(A[Z_{M}], V^{N′_{0}}) → Hom_{A[Y]}(A[Z_{M}], V^{N_{0}})

and hence an M-equivariant map

Hom_{A[(Z_{M}^{+} )′]}(A[Z_{M}], V^{N′_{0}}) → Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}),

which we denote by ı_{N_{0},N′_{0},z}. Now define

ı_{N_{0},N′_{0}} := z^{−1}ı_{N_{0},N′_{0},z} : Hom_{A[(Z_{M}^{+} )′]}(A[Z_{M}], V^{N′_{0}}) → Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}).

3.1.12. Proposition. (1) The map ı_{N_{0},N′_{0}} is well-defined independently of the

choice of element z ∈ Z_{M} such that zN′_{0}z^{−1} ⊂ N_{0}.

(2) ı_{N_{0},N′_{0}} is an M-equivariant isomorphism.

(3) If P′′_{0} is another choice of compact open subgroup of P, giving rise to N′′_{0}, etc., then ı_{N_{0},N′_{0}}ı_{N′_{0},N′′_{0}} = ı_{N_{0},N′′_{0}}.

Proof. Since z^{−1} is central in M, and since ı_{N_{0},N′_{0},z} is M-equivariant, the same is true of the map z^{−1}ı_{N_{0},N′_{0},z}. This proves the M-equivariance claim of part (2).

If P′′_{0} is as in part (3), and z′ ∈ Z_{M} is such that z′N′′_{0}(z′)^{−1} ⊂ N′_{0}, then part (2) of Lemma 3.1.11 shows that ı_{N_{0},N′_{0},z}ı_{N′_{0},N′′_{0},z′} = ı_{N_{0},N′′_{0},zz′}. Thus

z^{−1}ı_{N_{0},N′_{0},z}(z′)^{−1}ı_{N′_{0},N′′_{0},z′} = (zz′)^{−1}ı_{N_{0},N′′_{0},zz′}.

(Here we have also used the fact that ı_{N_{0},N′_{0},z} is M-equivariant, and so in particular Z_{M}-equivariant.) This proves part (3).

If we take P′′_{0} = P_{0} in part (3), then we find that ı_{N_{0},N_{0},zz′} is induced by h_{N_{0},zz′}, and thus gives the action of zz′ ∈ Z_{M}^{+} on Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}). Consequently (zz′)^{−1}ı_{N_{0},N_{0},zz′} acts as the identity on Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}), and we find that z^{−1}ı_{N_{0},N′_{0},z} and (z′)^{−1}ı_{N′_{0},N_{0},z′} are mutually inverse isomorphisms. This completes the proof of claim (2). Also, since z and z′ are independent of one another, we find that the isomorphism z^{−1}ı_{N_{0},N′_{0},z} does not depend on the choice of z. This proves claim (1), and completes the proof of the proposition. □

The preceding result shows that Ord_{P} is well-defined, up to canonical isomorphism, independently of the choice of the open subgroup P_{0} of P. It is similarly independent of the choice of the Levi factor M of P. Indeed, if M′ is another Levi factor of P, then we may canonically identify M and M′ via the canonical isomorphisms M ⟵^{∼} P/N ⟶^{∼} M′. Furthermore, there is a uniquely determined element n ∈ N such that nMn^{−1} = M′, and the isomorphism induced by conjugation by n yields the same identification of M and M′. Via this canonical identification, we may and do regard any M′-representation equally well as an M-representation.

Given a choice of compact open subgroup P_{0} ⊂ P, write P′_{0} := nP_{0}n^{−1}, M′_{0} := nM_{0}n^{−1} = nMn^{−1} ⋂ P′_{0}, N′_{0} := nN_{0}n^{−1} = N ⋂ P′_{0}, (M′)^{+} := nM^{+}n^{−1} = {m′ ∈ M′ | m′N′_{0}(m′)^{−1} ⊂ N′_{0}}, and Z_{M′}^{+} := nZ_{M}^{+}n^{−1} = (M′)^{+} ⋂ Z_{M′}. We define the Hecke action of (M′)^{+} on V^{N′_{0}} in analogy to the Hecke action of M^{+} on V^{N_{0}}, and denote by

Ord′_{P} : Mod_{P}^{sm}(A) → Mod_{M}^{sm}(A)

the functor

V ↦ Hom_{A[Z_{M′}^{+} ]}(A[Z_{M′}], V^{N′_{0}})_{Z_{M′} −fin}.

(The analogue of Lemma 3.1.7 for P′_{0} and M′ shows that Hom_{A[Z_{M′}^{+} ]}(A[Z_{M′}], V^{N′_{0}}),

and so also Hom_{A[Z_{M′}^{+} ]}(A[Z_{M′}], V^{N′_{0}})_{Z_{M′} −fin}, is naturally an M′-representation,

and hence an M-representation.) Thus Ord′_{P} is the analogue of the functor Ord_{P}, but computed using the Levi factor M′ of P rather than M. (Note that we have already seen that this functor is well-defined independently of the particular choice of compact open subgroup P′_{0} used to compute it.)

Multiplication by n provides an isomorphism V^{N_{0}} ⟶^{∼} V^{N′_{0}}, which intertwines the operator h_{m} on V^{N_{0}} with the operator h_{nmn^{−1}} on V^{N′_{0}}, for each m ∈ M^{+}. Consequently, multiplication by n induces an M-equivariant isomorphism

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}) ⟶^{∼} Hom_{A[Z_{M′}^{+} ]}(A[Z_{M′}], V^{N′_{0}}),

and hence an M-equivariant isomorphism Ord_{P} (V) ⟶^{∼} Ord′_{P} (V). This shows that the functor Ord_{P} is well-defined, up to canonical isomorphism, independently of the choice of Levi factor of P.

3.2. Some elementary properties of Ord_{P}. In this subsection, we record some simple properties of the functor Ord_{P}. We maintain the notation of the preceding subsection (A, P, M, N, P_{0}, M_{0}, N_{0}, M^{+}, Z_{M}, Z_{M}^{+}, etc.), and begin with a lemma that records some useful facts related to the monoid Z_{M}^{+}.

3.2.1. Lemma. (1) The monoid Z_{M}^{+} contains a compact open subgroup of Z_{M}.

(2) If Z_{0} is a compact open subgroup of Z_{M}^{+}, then we may find a submonoid Z_{0} ⊂ Y ⊂ Z_{M}^{+} for which the quotient Y/Z_{0} is a finitely generated monoid, and such that Y generates Z_{M} as a group.

Proof. Since M_{0} is an open subgroup of M, the intersection M_{0} ⋂ Z_{M} is a compact open subgroup of Z_{M}, which is clearly contained in Z_{M}^{+}. This proves (1).

Suppose that Z_{0} is any compact open subgroup of Z_{M} contained in Z_{M}^{+}. We claim that Z_{M}/Z_{0} is a finitely generated abelian group. (The proof will be given in the following paragraph.) This quotient is generated by Z_{M}^{+}/Z_{0} (since, as we have already recalled, Z_{M}^{+} generates Z_{M} as a group). Thus we may find a finitely generated submonoid of Z_{M}^{+}/Z_{0} which generates Z_{M}/Z_{0}. The preimage of this submonoid in Z_{M}^{+} is a submonoid Y of Z_{M}^{+} satisfying the requirements of (2).

To see that Z_{M}/Z_{0} is finitely generated, let Z_{M}^{0} denote the (ℚ_{p}-points of the) connected component of (the algebraic group underlying) Z_{M}; so Z_{M}^{0} is (the group of ℚ_{p}-points of) a torus. If we write Z_{0}^{0} := Z_{0} ⋂ Z_{M}^{0}, then Z_{M}^{0}/Z_{0}^{0} has finite index in Z_{M}/Z_{0}, and so it suffices to show that the former group is finitely generated. Let S denote the maximal split quotient of Z_{M}^{0} (in the category of tori), and let S′ denote the kernel of the map Z_{M}^{0} → S. The torus S′ is then anisotropic, and hence (its groups of ℚ_{p}-points is) compact. The intersection S′ ⋂ Z_{0}^{0} is compact and open in the compact group S′, and hence of finite index. Thus, if we let S_{0} denote the image of Z_{0}^{0} in S, it suffices to show that S/S_{0} is finitely generated. Let S_{1} denote the maximal compact subgroup of S. Since S_{0} is of finite index in S_{1}, it suffices in turn to show that S/S_{1} is finitely generated. Since S is a product of copies of ℚ_{p}^{×}, this follows from the isomorphism ℚ_{p}^{×}/ℤ_{p}^{×} ⟶^{∼} ℤ. (We remark that in fact Z_{M} itself is topologically finitely generated. Indeed, this follows from what we have just proved, together with [5, Prop. 6.4.1].) □

3.2.2. Lemma. If {W_{i}}_{i∈I} is an inductive system of smooth Z_{M}^{+}-modules^{3}, then the natural map of A[Z_{M}]-modules

(3.2.3) lim_{\overrightarrow{i}} Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W_{i})_{Z_{M} −fin} → Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], lim_{\overrightarrow{i}} W_{i})_{Z_{M} −fin}

is an isomorphism.

Proof. Let ϕ ∈ Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W_{j})_{Z_{M} −fin} for some index j in the directed set I, and suppose that the image of ϕ under (3.2.3) vanishes, or equivalently, that the image of im(ϕ) under the map W_{j} → lim_{\overrightarrow{i}} W_{i} vanishes. Since Lemma 3.1.6 shows that

im(ϕ) is a finitely generated A-submodule of W_{j}, we may find an index k lying over j such that the image of im(ϕ) under the transition map W_{j} → W_{k} vanishes, or equivalently, such that the image of ϕ under the map Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W_{j})_{Z_{M} −fin} → Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W_{k})_{Z_{M} −fin} vanishes. This proves the injectivity of (3.2.3).

Now let ϕ′ ∈ Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], lim_{\overrightarrow{i}} W_{i})_{Z_{M} −fin}. Lemma 3.1.6 shows that im(ϕ′)

is a finitely generated and Z_{M}^{+}-invariant A-submodule of lim_{\overrightarrow{i}} W_{i}. Thus we may find

a finitely generated A-module U ⊂ W_{j} that maps isomorphically onto im(ϕ′), for some sufficiently large index j. Let Z_{0} ⊂ Z_{M}^{+} be a compact open subgroup that fixes U elementwise, so that the Z_{M}^{+}-action on U factors through Z_{M}^{+}/Z_{0}. Fix a submonoid Y of Z_{M}^{+} satisfying the conditions of Lemma 3.2.1 (2).

If we choose k over j large enough, then, as the monoid ring A[Y/Z_{0}] is Noetherian (since Y/Z_{0} is finitely generated), the image of the finitely generated A[Y/Z_{0}]- module A[Y/Z_{0}]U in W_{k} will map isomorphically onto its image in lim_{\overrightarrow{i}} W_{i}, which is

to say, onto im(ϕ′). Consequently, if we choose j large enough, then we may choose U to be an A[Y]-submodule of W_{j} that maps isomorphically onto im(ϕ′). We may then lift ϕ′ to an element

3I.e., smooth under the action of a compact open subgroup of Z_{M}^{+} ; such a subgroup exists, by Lemma 3.2.1 (1).

ϕ ∈ Hom_{A[Y]}(A[Z_{M}], U)_{Z_{M} −fin} ⊂ Hom_{A[Y]}(A[Z_{M}], W_{j})_{Z_{M} −fin}.

Since Y generates Z_{M} as a group, the natural inclusion

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W_{j})_{Z_{M} −fin} ⊂ Hom_{A[Y]}(A[Z_{M}], W_{j})_{Z_{M} −fin}

is an isomorphism. Thus ϕ is in fact an element of Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W_{j})_{Z_{M} −fin}, which maps to the given element ϕ′. This shows that (3.2.3) is surjective. □

3.2.4. Proposition. Ord_{P} : Mod_{P}^{sm}(A) → Mod_{M}^{sm}(A) is left-exact and additive, and commutes with inductive limits.

Proof. The functors V ↦ V^{N_{0}} and W ↦ Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], W)_{Z_{M} −fin} (on the categories of P -representations and M^{+}-modules respectively) are left-exact and additive, and (taking into account Lemma 3.2.2) commute with inductive limits. Thus Ord_{P} also has these properties. □

3.3. Preservation of admissibility. If V is an object of Mod_{G}^{sm}(A), then we may also regard it as an object of Mod_{P}^{sm}(A), and so define Ord_{P} (V), an object of Mod_{M}^{sm}(A). The main result of this subsection is Theorem 3.3.3 below, which shows that if V is an admissible smooth G-representation, then Ord_{P} (V) is an admissible smooth M-representation.

We begin with a key lemma, which allows us to gain some control over the Hecke Z_{M}^{+}-action on V^{N_{0}}. Recall that if I is a compact open subgroup of G, then we say that I admits an Iwahori decomposition with respect to P and \overline{P} if the product map

(I ⋂ \overline{N}) × (I ⋂ M) × (I ⋂ N) → I

is a bijection. Note that, by applying the bijection g ↦ g^{−1} from I to itself, we find that the product map

(I ⋂ N) × (I ⋂ M) × (I ⋂ \overline{N}) → I

is also a bijection.

3.3.1. Remark. Note that the product map induces an injection

\overline{N} × M × N ↪ G.

Thus the following pair of conditions is equivalent to I admitting an Iwahori decomposition:

(1) I ⊂ \overline{N}MN.

(2) If \overline{n}mn ∈ I, with \overline{n} ∈ \overline{N}, m ∈ M, n ∈ N, then each of \overline{n}, m, n lies in I. From this it follows immediately that if I and I′ are two subgroups, both ad-

mitting an Iwahori decomposition, then the same is true of I ∩ I′, i.e. I ∩ I′ = (I ∩ I′ ∩ \overline{N}) × (I ∩ I′ ∩ M) × (I ∩ I′ ∩ N).

3.3.2. Lemma. Let I_{0} and I_{1} be two compact open subgroups of G, both of which admit an Iwahori decomposition with respect to P and \overline{P}. Suppose furthermore that I_{1} ⋂ \overline{N} ⊂ I_{0} ⋂ \overline{N}, that I_{1} ⋂ M = I_{0} ⋂ M, and that I_{1} ⋂ N = I_{0} ⋂ N = N_{0}. If z_{0} ∈ Z_{M}^{+} is such that (I_{0} ⋂ \overline{N}) ⊂ z_{0}(I_{1} ⋂ \overline{N})z_{0}^{−1}, then h_{N_{0},z_{0}}(V^{I_{1}}) ⊂ V^{I_{0}}.

Proof. To ease notation, write M_{1} := I_{1} ⋂ M = I_{0} ⋂ M, and \overline{N}_{i} := I_{i} ⋂ \overline{N}, for i = 0, 1. Thus I_{0} = \overline{N_{0}}M_{1}N_{0}, while I_{1} = \overline{N}_{1}M_{1}N_{0}. Also, since z_{0} centralizes M, we see that z_{0}I_{1}z_{0}^{−1} = z_{0}\overline{N}_{1}z_{0}^{−1}M_{1}z_{0}N_{0}z_{0}^{−1}, and so our assumption that z_{0}\overline{N}_{1}z_{0}^{−1} ⊃ \overline{N}_{0} implies that z_{0}I_{1}z_{0}^{−1} ⋂ I_{0} = \overline{N}_{0}M_{1}z_{0}N_{0}z_{0}^{−1}.

Evidently z_{0}I_{1}z_{0}^{−1} ⊃ z_{0}I_{1}z_{0}^{−1} ⋂ I_{0}, and so z_{0}V^{I_{1}} = V^{z_{0}I_{1}z_{0}^{−1}} ⊂ V^{z_{0}I_{1}z_{0}^{−1} ⋂ I_{0}}. Thus to prove the lemma, it suffices to show that h_{N_{0},z_{0}N_{0}z_{0}^{−1}}(V^{z_{0}I_{1}z_{0}^{−1} ⋂ I_{0}}) ⊂ V^{I_{0}}.

Since M_{1} normalizes both N_{0} and z_{0}N_{0}z_{0}^{−1}, we see that h_{N_{0},z_{0}N_{0}z_{0}^{−1}}(V^{z_{0}I_{1}z_{0}^{−1} ⋂ I_{0}}) is invariant under M_{1} and N_{0}. Thus it suffices to show that it is furthermore invariant under \overline{N}_{0}.

Let \overline{n} ∈ \overline{N}_{0}, and v ∈ V^{z_{0}I_{1}z_{0}^{−1} ⋂ I_{0}}. Fix a set \{n_{l}\} of coset representatives for N_{0}/z_{0}N_{0}z_{0}^{−1}. Then

\overline{n}h_{N_{0},z_{0}N_{0}z_{0}^{−1}}v = \overline{n} ∑_{l} n_{l}v = ∑_{l} \overline{n}n_{l}v = ∑_{l} n′_{l}\overline{p}_{l}v = ∑_{l} n′_{l}v,

where we have used the Iwahori decomposition of I_{0} to rewrite each product \overline{n}n_{l} in the form n′_{l}\overline{p}_{l}, with n′_{l} ∈ N_{0} and \overline{p}_{l} ∈ \overline{N}_{0}M_{1}. We claim that the n′_{l} again form a set of coset representatives for N_{0}/z_{0}N_{0}z_{0}^{−1}. Given this, we find that

\overline{n}h_{N_{0},z_{0}N_{0}z_{0}^{−1}}v = h_{N_{0},z_{0}N_{0}z_{0}^{−1}}v,

and thus that V^{z_{0}I_{1}z_{0}^{−1} ⋂ I_{0}} is fixed by \overline{N}_{0}, as required.

Suppose now that (n′_{l})^{−1}n′_{l′} ∈ z_{0}N_{0}z_{0}^{−1} for some l ≠ l′. Then

n_{l}^{−1}n_{l′} = \overline{p}_{l}^{−1}(n′_{l})^{−1}n′_{l′} \overline{p}_{l′},

and the right hand side of this equation is a product of elements all lying in z_{0}I_{1}z_{0}^{−1} ⋂ I_{0}. Since the left hand side lies in N, we see from the Iwahori decomposition of z_{0}I_{1}z_{0}^{−1} ⋂ I_{0} that in fact n_{l}^{−1}n_{l′} ∈ z_{0}N_{0}z_{0}^{−1}, contradicting the fact that they are distinct coset representatives for N_{0}/z_{0}N_{0}z_{0}^{−1}. Thus {n′_{l}} is indeed a set of distinct coset representatives for N_{0}/z_{0}N_{0}z_{0}^{−1}, and the lemma is proved. □

We now choose a cofinal sequence {I_{i}}_{i≥0} of compact open subgroups of G, with each I_{i} normal in I_{0}, and such that each I_{i} admits an Iwahori decomposition with respect to P and \overline{P}. We write M_{i} := I_{i} ⋂ M, N_{i} := I_{i} ⋂ N, and \overline{N_{i}} := I_{i} ⋂ \overline{N}, for each i ≥ 0.

For each i ≥ j ≥ 0, we write I_{i,j} := \overline{N}_{i}M_{j}N_{0}. Each I_{i,j} is a compact open subgroup of I_{0} (since we may rewrite it as I_{i,j} = I_{i}M_{j}N_{0}, noting that M_{j}N_{0} is a subgroup of M_{0}N_{0}, and thus of I_{0}, and so normalizes I_{i}). The bijection

\overline{N}_{i} × M_{j} × N_{0} ⟶^{∼} I_{i,j}

provides an Iwahori decomposition of I_{i,j}.

3.3.3. Theorem. If V is an admissible smooth representation of G over A, then Ord_{P} (V) is an admissible smooth representation of M, and the canonical lifting of Definition 3.1.10 is an embedding.

Proof. Since V is smooth by assumption, it is the inductive limit of its submodules V [𝔪^{i}] (i ≥ 0). Since Ord_{P} is left-exact and preserves inductive limits (Proposition 3.2.4), the embedding Ord_{P} (V [𝔪^{i}]) ↪ Ord_{P} (V)[𝔪^{i}] is an isomorphism, and Ord_{P} (V) is the inductive limit of its submodules Ord_{P} (V)[𝔪^{i}]. It thus suffices to prove the theorem with V replaced by V [𝔪^{i}] for some i ≥ 0. We must show that, for each j ≥ 0, Ord_{P} (V)^{M_{j}} is finitely generated over A, and that the canonical lifting induces an embedding Ord_{P} (V)^{M_{j}} ↪ V^{N_{0}}. We will compute Ord_{P} (V) using the compact open subgroup P_{0} := M_{0}N_{0} of P. Observe that V^{M_{j}N_{0}} is invariant under the Hecke Z_{M}^{+}-action on V^{N_{0}} (since Z_{M}^{+} is central in M), and thus

Ord_{P} (V)^{M_{j}} = Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}})_{Z_{M} −fin}^{M_{j}} = Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{M_{j}N_{0}})_{Z_{M} −fin}.

Let ϕ ∈ Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{M_{j}N_{0}})_{Z_{M} −fin}. Lemma 3.1.6 (2) shows that im(ϕ) is a finitely generated Z_{M}^{+}-invariant A-module, and thus is contained in V^{I_{i,j}} for some sufficiently large i ≥ j. If we choose z_{0} ∈ Z_{M}^{+} so that z_{0}\overline{N}_{i}z_{0}^{−1} ⊃ \overline{N_{j}}, then Lemma 3.1.6 (3) shows that in fact im(ϕ) = h_{N_{0},z_{0}} im(ϕ) ⊂ h_{N_{0},z_{0}}(V^{I_{i,j}}) ⊂ V^{I_{j,j}} (the final inclusion holding by Lemma 3.3.2). Thus if we let U denote the maximal A-submodule of V^{I_{j,j}} that is invariant under the Hecke Z_{M}^{+}-action, then we see that in fact im(ϕ) ⊂ U. Consequently, Ord_{P} (V)^{M_{j}} ⊂ Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], U), and so is finitely generated over A, by Lemma 3.1.5. The same lemma shows that the canonical lifting induces an embedding Ord_{P} (V)^{M_{j}} ↪ U ⊂ V^{I_{j,j}} ⊂ V^{N_{0}}. □

3.4. Extension to the case of ϖ-adically complete representations. We begin by making the following definition.

3.4.1. Definition. If V is an object of Mod_{P}^{ϖ−cont}(A), then we define

Ord_{P} (V) := lim_{\overleftarrow{i}} Ord_{P} (V/ϖ^{i}V).

(Note that each of the quotients V/ϖ^{i}V is an object of Mod_{P}^{sm}(A/𝔪^{i}), by Remark 2.4.2, and hence its P -ordinary part is defined.)

We now establish some basic properties of the functor Ord_{P} on the category Mod_{P}^{ϖ−cont}(A). It will ease the notation, and lend itself to further applications, if we work in a more general situation. Thus, we let H denote an arbitrary p-adic analytic group, and let F denote any O-linear, O-module valued, left-exact functor on Mod_{H}^{sm}(A)[ϖ^{∞}]. We extend F to a functor on Mod_{H}^{ϖ−cont}(A) via the formula

(3.4.2)  F (V) := lim_{\overleftarrow{i}} F (V/ϖ^{i}V),

for any object V of Mod_{H}^{ϖ−cont}(A). For any such V, the O-module F (V) has a natural projective limit topology, obtained by endowing each of the terms F (V/ϖ^{i}V) appearing in the projective limit that defines F (V) with its discrete topology. We also note that F (V) is ϖ-adically complete and separated (since each term F (V/ϖ^{i}V) appearing in the projective limit is ϖ-adically discrete^{4}, hence ϖ-adicallly complete and separated, and a projective limit of ϖ-adically complete and separated O-modules is itself ϖ-adically complete and separated).

The following results establish some additional properties of F (V).

3.4.3. Proposition. (1) The natural map F (V [ϖ^{∞}]) → F (V)[ϖ^{∞}] is an iso-

morphism. In particular, F (V)[ϖ^{∞}] is of bounded exponent.

(2) There is a natural embedding F (V)_{fl} ↪ F (V_{fl}), whose cokernel is of bounded exponent.

(3) The projective limit topology and the ϖ-adic topology on F (V) coincide.

4I.e. the ϖ-adic topology on this module coincides with the discrete topology.

Proof. Suppose first that V is O-torsion free. Then for any i, j ≥ 0, the short exact sequence

0 ⟶ V/ϖ^{i}V ⟶^{ϖ^{j}} V/ϖ^{i+j}V ⟶ V/ϖ^{j}V ⟶ 0

gives rise to an exact sequence

0 ⟶ F (V/ϖ^{i}V) ⟶^{ϖ^{j}} F (V/ϖ^{i+j}V) ⟶ F (V/ϖ^{j}V).

Passing to the projective limit in i, we obtain an exact sequence

0 ⟶ F (V) ⟶^{ϖ^{j}} F (V) ⟶ F (V/ϖ^{j}V).

Thus F (V) is O-torsion free, which verifies conditions (1) and (2), and for any j ≥ 0, we obtain an embedding

F (V)/ϖ^{j}F (V) ↪ F (V/ϖ^{j}V).

This verifies condition (3), and completes the proof of the lemma for O-torsion free V.

Let us turn to the general case. Fix i ≥ 0 so that V [ϖ^{i}] = V [ϖ^{∞}]. For any j ≥ i, we have the diagram

\xymatrix{0 \ar[r] & V [ϖ^{i}] \ar[r] & V/ϖ^{j}V \ar[r] \ar[dr]^{ϖ^{i}} & V_{fl}/ϖ^{j}V_{fl} \ar[r] \ar[d] & 0 \\  &  &  & V/ϖ^{i+j}V}

whose top row is exact, which gives rise to the diagram

(3.4.4)  \xymatrix{0 \ar[r] & F (V [ϖ^{i}]) \ar[r] & F (V/ϖ^{j}V) \ar[r] \ar[dr]^{ϖ^{i}} & F (V_{fl}/ϖ^{j}V_{fl}) \ar[d] \\  &  &  & F (V/ϖ^{i+j}V)}

whose top row is again exact. Passing to the inductive limit over j, we obtain the diagram

\xymatrix{0 \ar[r] & F (V [ϖ^{i}]) \ar[r] & F (V) \ar[r] \ar[dr]^{ϖ^{i}} & F (V_{fl}) \ar[d] \\  &  &  & F (V)}

with an exact top row. The result of the first paragraph shows that F (V_{fl}) is O-torsion free, and so we see that F (V [ϖ^{i}]) maps isomorphically onto F (V)[ϖ^{∞}], verifying condition (1), and also that there is an induced injection F (V)_{fl} ↪ F (V_{fl}), whose cokernel is annihilated by ϖ^{i}, verifying condition (2).

We now derive (3). For any j ≥ 0, let K_{j} ⊂ F (V) denote the kernel of the natural map F (V) → F (V/ϖ^{j}V). The sequence {K_{j}}_{j≥0} forms a basis of neighbourhoods of 0 in the projective limit topology on F (V). Clearly ϖ^{j}F (V) ⊂ K_{j}.

Taking into account condition (1), which we have already proved, and the exactness of the top row of (3.4.4), we find that the map F (V)[ϖ^{∞}] → F (V/ϖ^{j}V) is injective if j ≥ i, and thus that K_{j} ⋂ F (V)[ϖ^{∞}] = 0. If we let K′_{j} denote the image of K_{j} under the surjection F (V) → F (V)_{fl}, it follows that, when j ≥ i, the induced surjection K_{j} → K′_{j} is in fact an isomorphism.

It follows from the results of the first paragraph of the proof that the kernel of the map F (V_{fl}) → F (V_{fl}/ϖ^{j}V_{fl}) is precisely ϖ^{j}F (V_{fl}), and thus that the image of K′_{j} under the injection F (V)_{fl} ↪ F (V_{fl}) of (2) is contained in ϖ^{j}F (V_{fl}). Since, as we have shown above, the cokernel of this injection is annihilated by ϖ^{i}, we find that ϖ^{i}K′_{j} ⊂ ϖ^{j}F (V)_{fl}, and thus (since F (V)_{fl} is O-torsion free) that K′_{j} ⊂ ϖ^{j−i}F (V)_{fl}, provided that j ≥ i.

Both K_{j} and ϖ^{j−i}F (V) are subsets of K_{j−i}, and we have just shown that the image of the first in K′_{j−i} is contained in the image of the second. If we now suppose that j ≥ 2i, so that the surjection K_{j−i} → K′_{j−i} is also an isomorphism, we conclude that K_{j} ⊂ ϖ^{j−i}F (V). Thus the two decreasing sequences of Osubmodules {K_{j}}_{j≥0} and {ϖ^{j}F (V)}_{j≥0} of F (V) do indeed define the same topology on F (V), proving (3). □

3.4.5. Corollary. If H_{1} and H_{2} are two p-adic analytic groups, if F : Mod_{H_{1}}^{sm}(A) → Mod_{H_{2}}^{sm}(A) is a left-exact functor, and if we extend F to a functor on Mod_{H_{1}}^{ϖ−cont}(A) via the formula (3.4.2), then the extended functor takes values in Mod_{H_{2}}^{ϖ−cont}(A). If in addition the restriction of F to Mod_{H_{1}}^{adm}(A) takes values in Mod_{H_{2}}^{adm}(A), then the restriction to Mod_{H_{1}}^{ϖ−adm}(A) of the extension of F takes values in Mod_{H_{2}}^{ϖ−adm}(A).

Proof. The preceding proposition implies, for any object V of Mod_{H_{1}}^{ϖ−cont}(A), that F (V) is ϖ-adically complete and separated and that F (V)[ϖ^{∞}] has bounded exponent, verifying conditions (1) and (2) of Definition 2.4.1 for F (V). We also conclude that the projective limit topology on F (V) coincides with the ϖ-adic topology. Consequently, for any i ≥ 0, we may find j ≥ 0 such that F (V)/ϖ^{i}F (V) is a subquotient of F (V/ϖ^{j}V). Since the latter representation is a smooth H_{2}- representation, so is the former, by Lemma 2.2.6. Remark 2.4.2 now shows that F (V) is an object of Mod_{H_{2}}^{ϖ−cont}(A).

Suppose now that F restricts to a functor Mod_{H_{1}}^{adm}(A) → Mod_{H_{2}}^{adm}(A). If V is an object of Mod_{H_{1}}^{ϖ−adm}(A), then V/ϖ^{j}V is an object of Mod_{H_{1}}^{adm}(A), for each j ≥ 0, and so F (V/ϖ^{j}V) is an object of Mod_{H_{2}}^{adm}(A). As shown in the preceding paragraph, for any i ≥ 0, the quotient F (V)/ϖ^{i}F (V) is a subquotient of F (V/ϖ^{j}V) for some j ≥ 0, and thus is also an object of Mod_{H_{2}}^{adm}(A), by Proposition 2.2.13. We conclude that F (V) is an object of Mod_{H_{2}}^{ϖ−adm}(A), as claimed. □

3.4.6. Proposition. Taking ordinary parts is a functor Ord_{P} : Mod_{P}^{ϖ−cont}(A) → Mod_{M}^{ϖ−cont}(A).

Proof. Since Ord_{P} is a left-exact functor on Mod_{P}^{sm}(A), this follows directly from Corollary 3.4.5, together with the fact that the definition of Ord_{P} on Mod_{G}^{ϖ−cont}(A) is made according to the formula (3.4.2). □

For each i ≥ 0, we have the canonical lifting Ord_{P} (V/ϖ^{i}V) → (V/ϖ^{i}V)^{N_{0}}. These are compatible in an evident way as we change i, and so passing to the projective limit, we obtain an M^{+}-equivariant map

(3.4.7)  Ord_{P} (V) → V^{N_{0}},

which we again refer to as the canonical lifting.

3.4.8. Theorem. The passage to ordinary parts gives rise to a functor Ord_{P} : Mod_{G}^{ϖ−adm}(A) → Mod_{M}^{ϖ−adm}(A). Furthermore, for any object V of Mod_{G}^{ϖ−adm}(A), the canonical lifting (3.4.7) is a closed embedding, when source and target are given their ϖ-adic topologies.

Proof. The first claim of the theorem follows from Corollary 3.4.5, together with Theorem 3.3.3. That same theorem shows that if V is an object of Mod_{G}^{ϖ−adm}(A), then the canonical lifting Ord_{P} (V/ϖ^{j}V) → (V/ϖ^{j}V)^{N_{0}} is an embedding, for any j ≥ 0. Passing to the projective limit over j, we find that the canonical lifting Ord_{P} (V) → V^{N_{0}} is a topological embedding, when source and target are given their corresponding projective limit topologies. Proposition 3.4.3 (applied to each of the left-exact functors Ord_{P} and (–)^{N_{0}}) implies that these topologies coincide with the ϖ-adic topologies on source and target, completing the proof of the theorem. □

4. Adjunction formulas

4.1. Parabolic induction. Let A denote an object of Comp(O), with maximal ideal 𝔪. If U is an object of Mod_{M}^{sm}(A), then we regard U as a \overline{P} -representation, via the projection \overline{P} → \overline{P} /\overline{N} ⟶^{∼} M, and define:

IndG ; \overline{P} U := \{f : G → U | f locally constant ,

f(\overline{p}g) = \overline{p}f(g) for all \overline{p} ∈ \overline{P} , g ∈ G\} .

The right regular action of G on functions induces a natural G-action on IndG ; \overline{P} U, making it an A[G]-module.

If U is an object of Mod_{M}^{ϖ−cont}(A), then again regarding U as a \overline{P} -representation, we define:

IndG ; \overline{P} U := \{f : G → U | f continuous when U is given its ϖ-adic topology ,

f(\overline{p}g) = \overline{p}f(g) for all \overline{p} ∈ \overline{P} , g ∈ G\} .

Again, the right regular action of G on functions induces a natural G-action on IndG ; \overline{P} U, making it an A[G]-module.

In order to establish some basic properties of the functor IndG ; \overline{P}, it will be convenient to choose a continuous section σ : \overline{P} \G → G to the projection G → \overline{P} \G. (Such a section exists, since G is a locally trivial \overline{P} -bundle over \overline{P} \G, and hence we may find a finite cover of \overline{P} \G by disjoint open and closed sets over which this bundle is trivialized.) If we write C^{sm}(\overline{P} \G, U) (resp. C(\overline{P} \G, U)) to denote the space of locally constant (resp. continuous) U-valued functions on \overline{P} \G, for an object U of Mod_{G}^{sm}(A) (resp. an object U of Mod_{G}^{ϖ−cont}(A), equipped with its ϖ-adic topology), then pulling back functions along σ induces natural A-linear isomorphisms

(4.1.1)  IndG ; \overline{P} U ⟶^{∼} C^{sm}(\overline{P} \G, U), for any object U of Mod_{M}^{sm}(A),

and

(4.1.2)  IndG ; \overline{P} U ⟶^{∼} C(\overline{P} \G, U), for any object U of Mod_{M}^{ϖ−cont}(A).

4.1.3. Lemma. If U is an object of Mod_{M}^{ϖ−cont}(A), then for each i ≥ 0, there is a natural isomorphism

(IndG ; \overline{P} U)/ϖ^{i}(IndG ; \overline{P} U) ⟶^{∼} IndG ; \overline{P}(U/ϖ^{i}U).

Proof. Taking into account the isomorphism (4.1.2), it suffices to show that

C(X, U)/ϖ^{i} C(X, U) ⟶^{∼} C(X, U/ϖ^{i}U)

for any profinite space X. For any j ≥ i, we have the pair of short exact sequences

0 ⟶ (ϖ^{j−i}U + U[ϖ^{i}])/ϖ^{j}U ⟶ U/ϖ^{j}U ⟶^{ϖ^{i}} ϖ^{i}U/ϖ^{j}U ⟶ 0

and

0 → ϖ^{i}U/ϖ^{j}U → U/ϖ^{j}U → U/ϖ^{i}U → 0.

Note that the projective systems {(ϖ^{j−i}U + U[ϖ^{i}])/ϖ^{j}U}_{j≥i} and {ϖ^{i}U/ϖ^{j}U}_{j≥i} both satisfy the Mittag–Leffler condition. Indeed, for the latter projective system, the transition maps are even surjective. For the former projective system, denoting the jth term in the system by M_{j}, the image of the map M_{k} → M_{j} (for k ≥ j ≥ i) is (ϖ^{min(k−i,j)}U + U[ϖ^{i}])/ϖ^{j}U. If k ≥ j + i, then k − i ≥ j, and so this image simplifies to (ϖ^{j}U + U[ϖ^{i}])/ϖ^{j}U, which is independent of k.

Upon applying the (manifestly exact) functor C^{sm}(X, –), we obtain the short exact sequences

0 ⟶ C^{sm}(X, (ϖ^{j−i}U + U[ϖ^{i}])/ϖ^{j}U) ⟶ C^{sm}(X, U/ϖ^{j}U)

⟶^{ϖ^{i}} C^{sm}(X, ϖ^{i}U/ϖ^{j}U) ⟶ 0

and

0 → C^{sm}(X, ϖ^{i}U/ϖ^{j}U) → C^{sm}(X, U/ϖ^{j}U) → C^{sm}(X, U/ϖ^{i}U) → 0.

Furthermore, the two projective systems {C^{sm}(X, (ϖ^{j−i}U + U[ϖ^{i}])/ϖ^{j}U)}_{j≥i} and {C^{sm}(X, ϖ^{i}U/ϖ^{j}U)}_{j≥i} again satisfy the Mittag–Leffler condition, since C^{sm}(X, –) is exact. Thus, passing to the projective limit in j, we obtain short exact sequences

0 ⟶ C(X, U[ϖ^{i}]) ⟶ C(X, U) ⟶^{ϖ^{i}} C(X, ϖ^{i}U) ⟶ 0

and

0 → C(X, ϖ^{i}U) → C(X, U) → C(X, U/ϖ^{i}U) → 0.

(The exactness of the second sequence also follows directly from the discreteness of U/ϖ^{i}U.) Putting these together yields the exact sequence

0 ⟶ C(X, U[ϖ^{i}]) ⟶ C(X, U) ⟶^{ϖ^{i}} C(X, U) ⟶ C(X, U/ϖ^{i}U) ⟶ 0,

proving the lemma. □

4.1.4. Lemma. For objects U of Mod_{M}^{sm}(A), the formation of IndG ; \overline{P} U commutes with the formation of inductive limits.

Proof. Taking into account the isomorphism (4.1.1), it suffices to show that the formation of C^{sm}(X, U) commutes with the formation of inductive limits for any profinite space X. However, this is clear, as any element of C^{sm}(X, U) assumes only finitely many different values in U (since X is compact). □

4.1.5. Proposition. Parabolic induction U ↦ IndG ; \overline{P} U gives rise to exact functors Mod_{M}^{sm}(A) → Mod_{G}^{sm}(A) and Mod_{M}^{ϖ−cont}(A) → Mod_{G}^{ϖ−cont}(A).

Proof. We first consider the case when U is smooth. As already noted in the proof of Lemma 4.1.3, the formation of locally constant functions is manifestly an exact functor. It follows from the isomorphism (4.1.1) that the formation of IndG ; \overline{P} U is exact in U. We must show that it furthermore yields an object in Mod_{G}^{sm}(A). Fix a compact open subgroup H of G. Since \overline{P} \G is compact, it is the union of finitely many H-orbits, say \overline{P} g_{1}H, …, \overline{P} g_{r}H. Restricting functions from G to ∐_{i=1}^{r} g_{i}H thus induces the first of two isomorphisms in the following sequence of maps:

(4.1.6)

IndG ; \overline{P} U

⟶^{∼} ⨁_{i=1}^{r}\{f ∈ C^{sm}(g_{i}H, U) | f(\overline{p}g_{i}h) = \overline{p}f(g_{i}h) for all \overline{p} ∈ \overline{P} ⋂ g_{i}Hg_{i}^{−1}\}

⟶^{∼} ⨁_{i=1}^{r}\{f ∈ C^{sm}(H, U) | f(g_{i}^{−1}\overline{p}g_{i}h) = \overline{p}f(h) for all \overline{p} ∈ \overline{P} ⋂ g_{i}Hg_{i}^{−1}\}

↪ ⨁_{i=1}^{r} C^{sm}(H, U),

in which the second isomorphism is induced by left translation by g_{i}^{−1} on the ith summand, and the third map is the obvious embedding. If we equip each of the summands with the H-action given by right translation, then these maps become H-equivariant. The right regular H-action on C^{sm}(H, U) makes this space an object of Mod_{H}^{sm}(A), since any element of C^{sm}(H, U) is uniformly locally constant (since H is compact) and assumes only finitely many values in U. Thus IndG ; \overline{P} U embeds into a finite direct sum of smooth H-representations, and so is itself a smooth H-representation.

We now consider the case of ϖ-adically continuous representations. If 0 → U_{1} → U_{2} → U_{3} → 0 is a short exact sequence in Mod_{G}^{ϖ−cont}(A), then for any i ≥ 0 we obtain the short exact sequence 0 → U_{1}/(U_{1} ⋂ ϖ^{i}U_{2}) → U_{2}/ϖ^{i}U_{2} → U_{3}/ϖ^{i}U_{3} → 0, which in turn yields a short exact sequence

0 → C^{sm}(\overline{P} \G, U_{1}/(U_{1} ⋂ ϖ^{i}U_{2})) → C^{sm}(\overline{P} \G, U_{2}/ϖ^{i}U_{2})

→ C^{sm}(\overline{P} \G, U_{3}/ϖ^{i}U_{3}) → 0.

Since the projective system {U_{1}/(U_{1} ⋂ ϖ^{i}U_{2})}_{i≥0} has surjective transition maps, so does the projective system \{C^{sm}(\overline{P} \G, U_{1}/(U_{1} ⋂ ϖ^{i}U_{2}))\}_{i≥0}. Thus, passing to the projective limit in i, we obtain a short exact sequence

0 → C(\overline{P} \G, U_{1}) → C(\overline{P} \G, U_{2}) → C(\overline{P} \G, U_{3}) → 0.

(Here we have used Proposition 2.4.4, which shows that the topology on U_{1} induced by the decreasing sequence of O-submodules U_{1} ⋂ ϖ^{i}U_{2} coincides with the ϖ-adic topology on U_{1}.) Taking into account the isomorphism (4.1.2), we see that IndG ; \overline{P} induces an exact functor on Mod_{M}^{ϖ−cont}(A). It remains to show that it takes values in Mod_{G}^{ϖ−cont}(A).

If U is any object of Mod_{M}^{ϖ−cont}(A), then the space IndG ; \overline{P} U is ϖ-adically complete and separated, since the isomorphism (4.1.2) identifies it with a space of continuous functions with values in the ϖ-adically complete and separated space U.

Lemma 4.1.3 then shows that there is a natural isomorphism

IndG ; \overline{P} U ⟶^{∼} lim_{\overleftarrow{i}} IndG ; \overline{P}(U/ϖ^{i}U),

and thus that the functor IndG ; \overline{P} on Mod_{M}^{ϖ−cont}(A) is naturally isomorphic to the functor on Mod_{M}^{ϖ−cont}(A) obtained from the functor IndG ; \overline{P} on Mod_{M}^{sm}(A) according to formula (3.4.2). Corollary 3.4.5 now implies that IndG ; \overline{P} takes values in Mod_{G}^{ϖ−cont}(A). □

4.1.7. Proposition. If U is an admissible smooth (resp. locally admissible smooth, resp. ϖ-adically admissible) A[M]-module, then IndG ; \overline{P} U is an admissible smooth (resp. locally admissible smooth, resp. ϖ-adically admissible) A[G]-module.

Proof. In light of Proposition 4.1.5 and Lemmas 4.1.3 and 4.1.4, it suffices to consider the case when U is an object of Mod_{G}^{adm}(A). Let H be a compact open subgroup of G, and let j ≥ 0. As in the proof of Proposition 4.1.5, let \{\overline{P} g_{1}, …, \overline{P} g_{r}\} ⊂ G be a set of representatives for the finitely many orbits of H on \overline{P} \G. If we write M_{i} := M ∩ g_{i}Hg_{i}^{−1}, for i = 1, …, r, then the composition of the first two maps of (4.1.6) induces an embedding

(IndG ; \overline{P} U)^{H}[𝔪^{j}] ↪ ⨁_{i=1}^{r} U^{M_{i}}[𝔪^{j}].

Since each M_{i} is an open subgroup of M, and since U is assumed admissible, we see that the target of this embedding is finitely generated over A, and thus so is the source. This proves the proposition. □

Suppose now that U is an object of Mod_{M}^{sm}(G), so that elements of IndG ; \overline{P} U are locally constant. If f is such an element, then its support, which is an open and closed subset of G, is invariant under left translation by \overline{P}, and so corresponds to a compact open subset of the quotient \overline{P} \G. We will thus frequently speak of the support of f as a subset of \overline{P} \G, rather than of G.

Composing the closed embedding N ↪ G with the quotient map G → \overline{P} \G, we obtain an open immersion

ȷ : N ↪ \overline{P} \G,

which identifies N with an open subset ȷ(N) of \overline{P} \G. Let (IndG ; \overline{P} U)(ȷ(N)) denote the A-submodule of IndG ; \overline{P} U consisting of elements whose support (thought of as a compact open subset of \overline{P} \G) is contained in ȷ(N). Pulling back such a function to N via ȷ, we obtain a locally constant compactly supported function N → U. Thus, if we denote C_{c}^{sm}(N, U) the A-module of all such functions, then we have a map of A-modules

(4.1.8)  (IndG ; \overline{P} U)(ȷ(N)) → C_{c}^{sm}(N, U).

4.1.9. Lemma. The map (4.1.8) is an isomorphism.

Proof. Clearly this map is injective. On the other hand, given a function f ∈ C_{c}^{sm}(N, U), we may extend it to a function on \overline{P} N via the formula f(\overline{p}n) = \overline{p}f(n), and then extend it by zero to a function on G. Since f is compactly supported, this extension is locally constant, and yields an element of (IndG ; \overline{P} U)(ȷ(N)) that maps to f under (4.1.8). Thus (4.1.8) is also surjective. □

We denote the inverse isomorphism to (4.1.8) by ȷ_{∗}. Thus

ȷ_{∗} : C_{c}^{sm}(N, U) ⟶^{∼} (IndG ; \overline{P} U)(ȷ(N)).

The translation action of P on \overline{P} \G leaves its open subset ȷ(N) invariant. Concretely, we see that if mn ∈ MN = P, and n′ ∈ N, then

ȷ(n′)mn = ȷ(m^{−1}n′mn)

(where the left side denotes the translation action of mn on the element ȷ(n′) ∈ \overline{P} \G). Thus (IndG ; \overline{P} U)(ȷ(N)) is P -invariant. We may transport this P -action via the isomorphism of Lemma 4.1.9 to a P -action on C_{c}^{sm}(N, U). Of course, this P - action also admits a direct description, as follows:

(mnf)(n′) = mf(m^{−1}n′mn), for m ∈ M, n, n′ ∈ N, and f ∈ C_{c}^{sm}(N, U).

There is a natural isomorphism of A[P]-modules

C_{c}^{sm}(N, A) ⊗_{A} U ⟶^{∼} C_{c}^{sm}(N, U),

if we equip the left hand side with the tensor product P -action, where P acts on C_{c}^{sm}(N, A) via the above description (taking the M-action on the coefficients A to be trivial), and on U through its quotient M.

4.2. A simple adjunction formula. Fix a compact open subgroup P_{0} of P, and define M_{0}, N_{0}, M^{+}, etc., as in Subsection 3.1. We regard C_{c}^{sm}(N, A) as an A[P]- module via the formula of the preceding subsection (taking the M-action on A to be trivial). For any compact open subset Ψ ⊂ N, we denote by 1_{Ψ} ∈ C_{c}^{sm}(N, A) the characteristic function of Ψ (i.e. 1_{Ψ} is identically equal to 1 on Ψ, and vanishes elsewhere). If V is an object of Mod_{P}^{sm}(A), then (since M normalizes N), the space Hom_{A[N]}(C_{c}^{sm}(N, A), V) has a natural M-action.

4.2.1. Lemma. The map ev_{1_{N_{0}}} : Hom_{A[N]}(C_{c}^{sm}(N, A), V) → V^{N_{0}}, induced by evaluation at the function 1_{N_{0}} ∈ C_{c}^{sm}(N, A), is M^{+}-equivariant, if we equip the target with its Hecke M^{+}-action.

Proof. This is a simple calculation. □

Since the source of ev_{1_{N_{0}}} is an A[M]-module, it induces an M-equivariant map

(4.2.2) Hom_{A[N]}(C_{c}^{sm}(N, A), V) → Hom_{A[M^{+}]}(A[M], V^{N_{0}})

( ⟶^{∼} Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}) ),

where the isomorphism, which was observed in the proof of Lemma 3.1.7 (1), follows from [6, Prop. 3.3.6].

4.2.3. Proposition. The map (4.2.2) is an isomorphism.

Proof. Given ϕ ∈ Hom_{A[N]}(C_{c}^{sm}(N, A), V), let ϕ̃ ∈ Hom_{A[M^{+}]}(A[M], V^{N_{0}}) denote the image of ϕ under (4.2.2). If m ∈ M^{+}, then

(4.2.4)  ϕ̃(m) = (mϕ̃)(1) = (mϕ)(1_{N_{0}}) = ϕ(m1_{N_{0}}) = ϕ(1_{mN_{0}m^{−1}})

(where 1_{mN_{0}m^{−1}} denotes the characteristic function of mN_{0}m^{−1}). Since any element of C_{c}^{sm}(N, A) may be written in the form ∑_{i} a_{i}n_{i}1_{mN_{0}m^{−1}} for some finite collections {a_{i}} ⊂ A and {n_{i}} ⊂ N, and some m ∈ M^{+}, we see that ϕ is completely determined by the function ϕ̃, and thus (4.2.2) is injective. Conversely, given any ϕ̃ ∈ Hom_{A[M^{+}]}(A[M], V^{N_{0}}), we can define ϕ ∈ Hom_{A[N]}(C_{c}^{sm}(N, A), V) via the formula (4.2.4), i.e.

ϕ(f) := ∑_{i} a_{i}n_{i}ϕ̃(m),

for any function f = ∑_{i} a_{i}n_{i}1_{mN_{0}m^{−1}} ∈ C_{c}^{sm}(N, A); the M^{+}-equivariance of ϕ̃ ensures that ϕ(f) is well-defined, independently of the choice of representation of f as a sum of this form. Thus (4.2.2) is also surjective. □

4.2.5. Remark. Since the source of (4.2.2) is independent of the choice of N_{0}, this isomorphism yields another proof of Proposition 3.1.12.

4.2.6. Proposition. If U and V are objects of Mod_{M}^{sm}(A) and Mod_{P}^{sm}(A), respectively, then there is a natural isomorphism

Hom_{A[P]}(C_{c}^{sm}(N, U), V ) ⟶^{∼} Hom_{A[M]}(U, Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}})).

Proof. We compute

Hom_{A[P]}(C_{c}^{sm}(N, U), V ) ⟶^{∼} Hom_{A[P]}(C_{c}^{sm}(N, A) ⊗_{A} U, V )

⟶^{∼} Hom_{A[P]}(U, Hom_{A}(C_{c}^{sm}(N, A), V))

⟶^{∼} Hom_{A[M]}(U, Hom_{A[N]}(C_{c}^{sm}(N, A), V))

⟶^{∼} Hom_{A[M]}(U, Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}})),

where the third isomorphism arises from the fact that the P -action on M factors through M = P/N, and the final isomorphism is provided by Proposition 4.2.3. □

4.2.7. Proposition. If U is a locally Z_{M}-finite smooth representation of M over A then there is a natural M-equivariant isomorphism U ⟶^{∼} Ord_{P} (C_{c}^{sm}(N, U)), characterized by the fact that its composite with the canonical lifting Ord_{P} (C_{c}^{sm}(N, U)) → C_{c}^{sm}(N, U)^{N_{0}} is given by mapping an element u ∈ U to the constant function u supported on N_{0}.

Proof. If we write U = ⋃_{i∈I} U_{i} as the directed union of finitely generated, Z_{M}- invariant, A-modules, then we may write

C_{c}^{sm}(N, U)^{N_{0}} ⟶^{∼} lim_{\overrightarrow{i∈I,z∈Z{+ \atop M}}} C^{sm}(z^{−1}N_{0}z, U_{i})^{N_{0}} ⟶^{∼} lim_{\overrightarrow{i∈I,z∈Z{+ \atop M}}} A[z^{−1}N_{0}z/N_{0}] ⊗_{A} U_{i},

where, in forming the inductive limits, we direct Z_{M}^{+} via the relation of divisibility. Lemma 3.2.2 yields an isomorphism

lim_{\overrightarrow{i,z}} Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], A[z^{−1}N_{0}z/N_{0}] ⊗_{A} U_{i})_{Z_{M} −fin} ⟶^{∼} Ord_{p}(C_{c}^{sm}(N, U)).

It is easily checked that evaluation at 1 induces an isomorphism

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], A[z^{−1}N_{0}z/N_{0}] ⊗_{A} U_{i})_{Z_{M} −fin} ⟶^{∼} U_{i},

where U_{i} is embedded into A[z^{−1}N_{0}z/N_{0}] ⊗_{A} U_{i} via u ↦ 1 ⊗ u. Thus we find that

U ⟶^{∼} lim_{\overrightarrow{i}} U_{i} ⟶^{∼} Ord_{p}(C_{c}^{sm}(N, U))

in the manner claimed. We leave it to the reader to verify that this isomorphism is M-equivariant. □

4.2.8. Corollary. If U is a locally Z_{M}-finite smooth representation of M over A, and if V is a smooth representation of P over A, then passing to ordinary parts induces a natural isomorphism

Hom_{A[P]}(C_{c}^{sm}(N, U), V ) ⟶^{∼} Hom_{A[M]}(U, Ord_{P} (V)).

Proof. This follows from the preceding propositions, together with the fact that, since U is locally Z_{M}-finite by assumption, the inclusion

Hom_{A[M]}(U, Ord_{P} (V)) ⊂ Hom_{A[M]}(U, Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}}))

is in fact an equality. □

4.3. Ordinary parts of parabolically induced representations. In this subsection we will strengthen Proposition 4.2.7 by computing the P -ordinary part of a representation parabolically induced from \overline{P}.

Fix a minimal parabolic subgroup P_{0} of G (defined over ℚ_{p}) contained in P, choose a Levi factor M_{0} of P_{0} contained in M, and let W denote the Weyl group of G with respect to M_{0}. We may choose a subset W_{P} ⊂ W so that G decomposes as the disjoint union of locally closed subsets

G = ∐_{w∈w_{P}} \overline{P} wP.

We may and do choose W_{P} so that it contains the identity 1 ∈ W, and also the longest element w_{ℓ} ∈ W. We define a partial ordering on W_{P} as follows: w ≻ w′ if and only if \overline{P} wP is contained in the closure of \overline{P} w′P. (See for instance the discussion at the beginning of [3, §6.3]; note, though, that the parabolic subgroups P_{Θ} and P_{Ω} considered there are both supposed to be attached to subsets Θ and Ω of positive simple roots. Thus to compare our situation with that of [3, §6.3], one should write P = P_{Ω} for some set of positive simple roots, as well as \overline{P} = w_{ℓ}^{−1}P w_{ℓ}, so that the above decomposition of G gets rewritten as G = ∐_{w∈w_{P}} P_{Ω}w_{ℓ}wP_{Ω}. In [3, §1.1] a subgroup W_{Ω} of W associated to Ω is defined, and a particular set of double-coset representatives [W_{Ω}\W/W_{Ω}] of the double quotient W_{Ω}\W/W_{Ω} is chosen. If w ∈ W, let [w] ∈ [W_{Ω}\W/W_{Ω}] denote the representative of W_{Ω}wW_{Ω}. The map w ↦ [w_{ℓ}w] then induces a bijection between the subset W_{P} of W and the set [W_{Ω}\W/W_{Ω}], which is order-reversing, when W_{P} is equipped with the order introduced above, and [W_{Ω}\W/W_{Ω}] is equipped with the order defined in [3, §6.3].)

For any w ∈ W_{P} , write

S_{w} := ∐_{w′∈W_{P} ,w≻w′} \overline{P} \\overline{P} w′P.

Then each S_{w} is an open subset of \overline{P} \G, S_{1} := ȷ(N), S_{w_{ℓ}} := \overline{P} \G, and S_{w′} ⊂ S_{w} if and only if w ≻ w′.

If U is a locally Z_{M}-finite smooth representation of M over A, then for each w ∈ W_{P} , we let (IndG ; \overline{P} U)(S_{w}) denote the subspace of IndG ; \overline{P} U consisting of functions whose support (when regarded as a subset of \overline{P} \G) lies in S_{w}. Since each S_{w} is a union of P -orbits in \overline{P} \G, each subspace (IndG ; \overline{P} U)(S_{w}) is P -invariant.

4.3.1. Lemma. If w ≻ w′ and w is an immediate successor of w′ in W_{P} (i.e. if w ≠ w′, and if there is no element w′′ ∈ W_{P} such that w ⪶ w′′ ⪶ w′), then

Ord_{P} ((IndG ; \overline{P} U)(S_{w})/(IndG ; \overline{P} U)(S_{w′})) = 0.

Proof. Restricting elements of (IndG ; \overline{P} U)(S_{w}) to \overline{P} wP induces an exact sequence of P -representations 0 → (IndG ; \overline{P} U)(S_{w′}) → (IndG ; \overline{P} U)(S_{w}) → C, where

C := \{f : \overline{P} wP → U | f(\overline{n}mwp) = mf(wp) for all \overline{n} ∈ \overline{N}, m ∈ M, p ∈ P,

and the support of f is compact modulo \overline{P} \}

is equipped with the right regular P -action. Since Ord_{P} is a left-exact functor, it suffices to show that Ord_{P} (C) = 0. Suppose that

ϕ ∈ Ord_{P} (C) := Hom_{A[Z_{M}]^{+}}(A[Z_{M}], C^{N_{0}})_{Z_{M} −fin}.

Lemma 3.1.6 (2) shows that im(ϕ) is a finitely generated A-submodule of C. In particular, we may find a compact open subset Ω of \overline{P} \\overline{P} wP such that the support of every element of im(ϕ), taken modulo \overline{P}, lies in Ω. Since Ω is compact, we may find z ∈ Z_{M}^{+} such that Ωz^{−1} ⊂ \overline{P} \\overline{P} wN_{0}. Since Lemma 3.1.6 (3) implies that

(4.3.2)  im(ϕ) = h_{N_{0},z} im(ϕ),

we see that in fact every element of im(ϕ) has its support contained in \overline{P} \\overline{P} wN_{0}. Since ϕ takes values in C^{N_{0}}, every element f of im(ϕ) is furthermore invariant under the right regular action of elements of N_{0}. Thus, if we write f(w) = u, then, when restricted to its support (i.e. to \overline{P} \\overline{P} wN_{0}), the function f is given by the formula

(4.3.3)  \overline{n}mwn_{0} ↦ mu.

We write f =: f_{u}.

If z ∈ Z_{M}^{+}, then h_{N_{0},z}f_{u} is again supported on \overline{P} \\overline{P} wN_{0}, and invariant under N_{0}, and so we may write h_{N_{0},z}f_{u} = f_{u′}, for some u′ ∈ U. To compute u′, we evaluate h_{N_{0},z}f_{u} at w:

u′ = (h_{N_{0},z}f)(w) = ∑_{n∈N_{0}/zN_{0}z^{−1}}(nzf_{u})(w) = ∑_{n∈N_{0}/zN_{0}z^{−1}} f_{u}(wnz).

Since f_{u} is supported on \overline{P} wN_{0}, we see (taking into account the formula (4.3.3)) that

f_{u}(wnz) = \begin{cases}0 if n ∉ w^{−1}\overline{N}w ⋂ N_{0} \\ (wzw^{−1})u if n ∈ w^{−1}\overline{N}w ⋂ N_{0}. \end{cases}

Thus

u′ = [(N_{0} ⋂ w^{−1}\overline{N}w) : (zN_{0}z^{−1} ⋂ w^{−1}\overline{N}w)] · (wzw^{−1})u.

Since w is a non-trivial element of W_{P} (since it is strictly greater than the element w′ ∈ W_{P}), the intersection N ⋂ w^{−1}\overline{N}w is non-trivial, and so the intersection N_{0} ⋂ w^{−1}\overline{N}w is a non-trivial pro-p group, which is topologically finitely generated, but of infinite order. As z ranges over the elements of Z_{M}^{+}, the groups zN_{0}z^{−1} range over a cofinal collection of open subgroups of N_{0}. Thus the intersections zN_{0}z^{−1} ⋂ w^{−1}\overline{N}w range over a cofinal collection of open subgroups of N_{0} ⋂ w^{−1}\overline{N}w, and so the index [(N_{0} ⋂ w^{−1}\overline{N}w) : (zN_{0}z^{−1} ⋂ w^{−1}\overline{N}w)] can be made an arbitrarily large power of p by choosing z ∈ Z_{M}^{+} appropriately. Since u is annihilated by some power of p, we conclude that u′ = 0. From (4.3.2) we conclude that in fact im(ϕ) = 0, and the lemma follows, since ϕ was an arbitrarily chosen element of Ord_{P} (C). □

4.3.4. Proposition. If U is a locally Z_{M}-finite smooth M-representation over A, then there is a natural isomorphism of M-representations U ⟶^{∼} Ord_{P} (IndG ; \overline{P} U).

Proof. It follows from the preceding lemma, together with the left-exactness of Ord_{P}, that the inclusion

(IndG ; \overline{P} U)(ȷ(N)) ⊂ IndG ; \overline{P} U

induces an isomorphism on P -ordinary parts. The proposition now follows from Proposition 4.2.7 and Lemma 4.1.9. □

4.3.5. Corollary. If U is an object of Mod_{M}^{ϖ−adm}(A), then there is a natural isomorphism of M-representations U ⟶^{∼} Ord_{P} (IndG ; \overline{P} U).

Proof. For each integer i ≥ 0, the preceding proposition, together with Lemma 2.3.4, yields an isomorphism

U/ϖ^{i}U ⟶^{∼} Ord_{P} (IndG ; \overline{P}(U/ϖ^{i}U)).

Passing to the projective limit over i, and taking into account Definition 3.4.1 and Lemma 4.1.3, we obtain the required isomorphism. □

4.4. The main adjunction formula. The main result of this subsection is Theorem 4.4.6, showing that IndG ; \overline{P} and Ord_{P} are adjoint functors.

Let U be a smooth, admissible representation of M over A, and let V be a smooth representation of G over A. Recall from Subsection 4.1 that ȷ_{∗} provides a P -equivariant isomorphism between C_{c}^{sm}(N, U) and (IndG ; \overline{P} U)(ȷ(N)).

If Ω is a compact open subset of \overline{P} \G, and f ∈ IndG ; \overline{P} U, then we will let f_{|Ω} denote the restriction of f to the preimage (under the canonical projection G → \overline{P} \G) of Ω in G, extended by zero to a function on G. The function f_{|Ω} again lies in IndG ; \overline{P} U, and the support of f_{|Ω}, thought of as a subset of \overline{P} \G, is equal to the intersection of Ω and the support of f. The reader may verify the following formula, which we will apply repeatedly: for any g ∈ G, we have that

(4.4.1)  (gf)_{|Ω} = g(f_{|Ωg}).

We will apply similar notation when considering elements of C_{c}^{sm}(N, U): i.e. if f ∈ C_{c}^{sm}(N, U), and if Ψ is a compact open subset of N, then we will let f_{|Ψ} denote the restriction of f to Ψ, extended by zero over N. We have the following formula, analogous to (4.4.1): if mn ∈ MN = P, then

(4.4.2)  (mnf)_{|Ψ} = mn(f_{|m^{−1}Ψmn}).

4.4.3. Lemma. Let ϕ ∈ Hom_{A[P]}((IndG ; \overline{P} U)(ȷ(N)), V ). If f ∈ (IndG ; \overline{P} U)(ȷ(N)) and g ∈ G are chosen so that gf also lies in (IndG ; \overline{P} U)(ȷ(N)), then ϕ(gf) = gϕ(f).

Proof. Let Ω denote the support of f, which by assumption is contained in ȷ(N), and write Ψ = ȷ^{−1}(Ω) ⊂ N. As in Subsection 3.3, choose a cofinal sequence {I_{i}}_{i≥0} of compact open subgroups of G, each admitting an Iwahori decomposition \overline{N}_{i} × M_{i} × N_{i} ⟶^{∼} I_{i}. Since Ω, and hence Ψ, is compact, we may choose i sufficiently large so that Ψ is a union of left cosets of N_{i}, i.e. so that N_{i}n ⊂ Ψ for every n ∈ Ψ. Write f = ȷ_{∗}f′, with f′ ∈ C_{c}^{sm}(N, U). Since f′ is a locally constant function, and in particular assumes only finitely many distinct values in U, enlarging i if necessary, and choosing i′ sufficiently large, we may furthermore assume that f′ is left-N_{i}- invariant^{5}, and takes values in U^{M_{i}}[𝔪^{i′}]. If n ∈ Ψ, we then see that (nf′)_{|N_{i}} is a constant U^{M_{i}}[𝔪^{i′}]-valued function on N_{i}.

5I.e. constant on left N_{i}-cosets.

We let X denote the subspace of C_{c}^{sm}(N, U) consisting of constant U^{M_{i}}[𝔪^{i′}]- valued functions on N_{i}, regarded as functions on N via extension by zero. The previous paragraph shows that (nf′)_{|N_{i}} ∈ X.

We claim that ȷ_{∗}(X) is elementwise-fixed by I_{i}. Note that ȷ_{∗}(X) equals the set of functions f_{u} : G → U which are supported on \overline{P} N_{i} = \overline{N}MN_{i}, and which on that set are defined by the formula f_{u}(\overline{n}mn) = mu, for \overline{n} ∈ \overline{N}, m ∈ M, and n ∈ N_{i}, where u ranges over the elements of U^{M_{i}}[𝔪^{i′}]. Note next (taking into account the Iwahori decomposition of I_{i}) that \overline{P} N_{i} = \overline{P} I_{i}, and thus that if F is any function on G supported on \overline{P} N_{i}, the same is true of gF, if g ∈ I_{i}. Furthermore, if g ∈ I_{i} and n ∈ N_{i} then the product ng again lies in I_{i}, and so (again taking into account the Iwahori decomposition of I_{i}) may be written in the form g = \overline{n}′m′n′, for some \overline{n}′ ∈ \overline{N}_{i}, m′ ∈ M_{i}, and n′ ∈ N_{i}. We thus compute that, for any \overline{n} ∈ \overline{N} and m ∈ M, that

(gf_{u})(\overline{n}mn) = f_{u}(\overline{n}mng) = f_{u}(\overline{n}m\overline{n}′m′n′) = f_{u}(\overline{n}(m\overline{n}′m^{−1})mm′n′)

= mm′u = mu = f_{u}(\overline{n}mn).

(Here the first equality holds by definition, since G acts by right translation; the second equality is just applying the Iwahori decomposition for ng given above; the third equality is clear; the fourth and sixth equalities follow from the definition of f_{u} as a function; and the fifth equality follows from the fact that u ∈ U^{M_{i}}.) Thus ȷ_{∗}(X) is indeed fixed elementwise by I_{i}.

Since U is assumed to be admissible smooth, the space U^{M_{i}}[𝔪^{i′}] is a finitely generated A-module, and thus the same is true of X and ȷ_{∗}(X). Thus ϕ(ȷ_{∗}(X)) is a finitely generated A-submodule of V, and we may choose j ≥ i so that I_{j} fixes ϕ(ȷ_{∗}(X)) elementwise.

The support of gf is equal to the translate Ωg^{−1} of Ω, which by assumption again lies in ȷ(N). Note that this translate is equal to the image under the natural projection \overline{P} \G → G of the translate Ψg^{−1} ⊂ Ng^{−1} ⊂ G. Since by assumption it lies ȷ(N), we see that Ψg^{−1} is a compact subset of \overline{P} N. Let \overline{Ψ} denote the projection of Ψg^{−1} onto \overline{N} (i.e. the projection onto the first factor in the source of the isomorphism \overline{N} × M × N ⟶^{∼} \overline{P} N). Choose an element z ∈ Z_{M} so that z\overline{Ψ}z^{−1} ⊂ \overline{N}_{j}, while z^{−1}N_{i}z ⊂ N_{i}.

Fix an n ∈ Ψ. Since Ψg^{−1} ⊂ \overline{Ψ}P, we may write gn^{−1} = p\overline{n} for some \overline{n} ∈ \overline{Ψ}^{−1} and p ∈ P. By virtue of our choice of z, we have z\overline{n}z^{−1} ∈ \overline{N}_{j}^{−1} = \overline{N}_{j}. We now compute:

(4.4.4) g(f_{|ȷ(N_{i})zn}) = gn^{−1}z^{−1}(znf)_{|ȷ(N_{i})} = p\overline{n}z^{−1}(znf)_{|ȷ(N_{i})}

= pz^{−1}z\overline{n}z^{−1}(znf)_{|ȷ(N_{i})} = pz^{−1}(znf)_{|ȷ(N_{i})}.

All but the final equality are trivial manipulations (taking into account the formula (4.4.1)). The final equality holds because (znf)_{|ȷ(N_{i})} lies in ȷ_{∗}(X) (as we will show in a moment), and so is fixed by the element z\overline{n}z^{−1} ∈ \overline{N}_{j} ⊂ \overline{N}_{i}. To see that (znf)_{|ȷ(N_{i})} ∈ ȷ_{∗}(X), note that since (nf′)_{|N_{i}} is a constant U^{M_{i}}[𝔪^{i′}]-valued function on N_{i}, an application of (4.4.2) shows that (znf′)_{|zN_{i}z^{−1}} = z((nf′)_{|N_{i}}) is a constant U^{M_{i}}[𝔪^{i′}]-valued function on zN_{i}z^{−1} ⊃ N_{i}. Thus (znf′)_{|N_{i}} is again a constant U^{M_{i}}[𝔪^{i′}]-valued on N_{i}, and so lies in X. Consequently (znf)_{|ȷ(N_{i})} = ȷ_{∗}((znf′)_{|N_{i}}) lies in ȷ_{∗}(X), as claimed.

Applying ϕ to (4.4.4), we compute that

(4.4.5) ϕ(g(f_{|ȷ(N_{i})zn})) = ϕ(pz^{−1}(znf)_{|ȷ(N_{i})}) = pz^{−1}ϕ((znf)_{|ȷ(N_{i})})

= pz^{−1}(z\overline{n}z^{−1})ϕ((znf)_{|ȷ(N_{i})}) = gn^{−1}z^{−1}ϕ((znf)_{|ȷ(N_{i})})

= gϕ(n^{−1}z^{−1}(znf)_{|ȷ(N_{i})}) = gϕ(f_{|ȷ(N_{i})zn}).

Here the first equality is given by (4.4.4), the second and fifth equalities follow from the P -equivariance of ϕ, and the third equality arises from the fact that ϕ((znf)_{|ȷ(N_{i})}) ∈ ϕ(ȷ_{∗}(X)) (by the discussion of the preceding paragraph), and hence is fixed by the element z\overline{n}z^{−1} ∈ \overline{N}_{j}. The fourth equality is a trivial manipulation, while the final equality is an application of (4.4.1).

Since z^{−1}N_{i}z ⊂ N_{i}, we may write Ψ as a finite disjoint union of left z^{−1}N_{i}z cosets, say Ψ = ∐_{l} z^{−1}N_{i}zn_{l}, and thus decompose Ω as a union of translates of ȷ(N_{i}), namely Ω = ∐_{l} ȷ(N_{i})zn_{l}. Accordingly, we may write f = ∑_{l} f_{|ȷ(N_{i})zn_{l}}, and then compute (taking into account (4.4.5)) that

ϕ(gf) = ∑_{l} ϕ(gf_{|ȷ(N_{i})zn_{l}}) = ∑_{l} gϕ(f_{|ȷ(N_{i})zn_{l}}) = gϕ(f).

This proves the lemma. □

We now prove our main result.

4.4.6. Theorem. Let A be an object of Comp(O). If U is an object of Mod_{M}^{adm}(A) (resp. Mod_{M}^{ϖ−adm}(A)) and V is an object of Mod_{G}^{sm}(A) (resp. Mod_{G}^{ϖ−cont}(A) ) then passage to ordinary parts induces an isomorphism

Hom_{A[G]}( IndG ; \overline{P} U, V ) ⟶^{∼} Hom_{A[M]}(U, Ord_{P} (V)).

Consequently, the functor Ord_{P} on Mod_{G}^{adm}(A) (resp. Mod_{G}^{ϖ−adm}(A)) is right adjoint to the functor U ↦ IndG ; \overline{P} U (which takes Mod_{M}^{adm}(A), resp. Mod_{M}^{ϖ−adm}(A), to Mod_{G}^{adm}(A), resp. Mod_{G}^{ϖ−adm}(A), by Proposition 4.1.7).

Proof. We first consider the case when U and V are smooth. Restricting maps from IndG ; \overline{P} U to (IndG ; \overline{P} U)(ȷ(N)) induces a map

(4.4.7) Hom_{A[G]}(IndG ; \overline{P} U, V ) → Hom_{A[P]}((IndG ; \overline{P} U)(ȷ(N)), V )

⟶^{∼} Hom_{A[G]}(A[G] ⊗_{A[P]} (IndG ; \overline{P} U)(ȷ(N)), V ).

The natural map

A[G] ⊗_{A[P]} (IndG ; \overline{P} U)(ȷ(N)) → IndG ; \overline{P} U

is surjective, as (IndG ; \overline{P} U)(ȷ(N)) generates IndG ; \overline{P} U over G, since the G-translates of ȷ(N) cover \overline{P} \G. Thus (4.4.7) is injective. We claim that it is in fact an isomorphism. To prove this, we have to show that any map of A[G]-modules

ϕ : A[G] ⊗_{A[P]} (IndG ; \overline{P} U)(ȷ(N)) → V,

in which the target is smooth, necessarily factors through the quotient IndG ; \overline{P} U of the source. In other words, we have to show that if g_{1}, …, g_{l} is a finite sequence of elements of G, and f_{1}, …, f_{l} is a finite sequence of elements of (IndG ; \overline{P} U)(ȷ(N)), such that

(4.4.8)  ∑_{i=1}^{l} g_{i}f_{i} = 0 in IndG ; \overline{P} U,

then

(4.4.9)  ∑_{i=1}^{l} g_{i}ϕ(f_{i}) =^{?} 0 in V.

It suffices to show that each point x of \overline{P} \G has a compact open neighbourhood Ω_{x} such that for any neighbourhood Ω′_{x} ⊂ Ω_{x} of x we have

(4.4.10)  ∑_{i=1}^{l} g_{i}ϕ(f_{i|Ω′_{x}g_{i}}) =^{?} 0.

Indeed, we can then partition \overline{P} \G into a finite disjoint union of such neighbourhoods, say \overline{P} \G = ∐_{j=1}^{s} Ω′_{x_{j}}, and writing

g_{i}f_{i} = ∑_{j=1}^{s}(g_{i}f_{i})_{|Ω′_{x_{j}}} = ∑_{j=1}^{s} g_{i}f_{i|Ω′_{x_{j}} g_{i}}

(where we have used (4.4.1)), so that

f_{i} = ∑_{j=1}^{s} f_{i|Ω′_{x_{j}} g_{i}},

we conclude that if (4.4.10) holds, then so does (4.4.9).

If x = \overline{P} g for some g ∈ G, then replacing (g_{1}, …, g_{l}) by (gg_{1}, …, gg_{l}), we see that it suffices to treat the case when x equals the identity coset, which is equal to the image ȷ(e) of the identity e of N under the open immersion ȷ : N ↪ \overline{P} \G. Let Ω_{e} be a compact open neighbourhood of ȷ(e) in ȷ(N), chosen so that Ω_{e}g_{i} ⊂ ȷ(N) for all i for which g_{i} ∈ \overline{P} N, and so that Ω_{e}g_{i} is disjoint from the support of f_{i} for all other i. It follows from (4.4.8) (and an application of (4.4.1)) that

∑_{i=1}^{l}(g_{i}f_{i})_{|Ω_{e}} = ∑_{i=1}^{l} g_{i}f_{i|Ω_{e}g_{i}} = 0 in (IndG ; \overline{P} U)(ȷ(N)).

Applying the map ϕ to the second of these equalities (which is an equation involving elements of (IndG ; \overline{P} U)(ȷ(N)), since f_{i|Ω_{e}g_{i}} = 0 if Ω_{e}g_{i} ⊄ ȷ(N), by virtue of our choice of Ω_{e}), and taking into account Lemma 4.4.3, we deduce that indeed ∑_{i=1}^{l} g_{i}ϕ(f_{i|Ω_{e}g_{i}}) = 0, and thus we have shown that (4.4.7) is an isomorphism. Now composing (4.4.7) with the isomorphism

Hom_{A[P]}((IndG ; \overline{P} U)(ȷ(N)), V ) ⟶^{∼} Hom_{A[P]}(C_{c}^{sm}(N, U), V )

induced by ȷ_{∗}, together with the isomorphism of Corollary 4.2.8, and taking into account Proposition 4.3.4 (note that these results apply, since Lemma 2.3.4 implies that U is locally Z_{M}-finite), completes the proof of the theorem in the case when U and V are smooth.

We now consider the case when U is ϖ-adically admissible and V is ϖ-adically continuous. There is a commutative diagram

\xymatrix{Hom_{A[G]}( IndG ; \overline{P} U, V ) \ar[r]^{∼} \ar[d] & lim_{\overleftarrow{i}} Hom_{(A/ϖ^{i}A)[G]}( IndG ; \overline{P}(U/ϖ^{i}U), V/ϖ^{i}V ) \ar[d] \\ Hom_{A[M]}(U, Ord_{P} (V)) \ar[r]^{∼} & lim_{\overleftarrow{i}} Hom_{(A/ϖ^{i}A)[M]}(U/ϖ^{i}U, Ord_{P} (V/ϖ^{i}V)),}

in which the vertical arrows are given by passing to ordinary parts (here we are taking into account Proposition 4.3.4 and Corollary 4.3.5), and the horizontal arrows arise from the isomorphism of Lemma 4.1.3 and Definition 3.4.1. By what we have already proved, the right hand vertical arrow is also an isomorphism. Thus so is the left hand vertical arrow. □

Appendix A. Functional analysis

In this appendix we establish some simple functional analytic results regarding (not necessarily finitely generated) modules over the ring O.

A.1. Lemma. If 0 → V_{1} → V_{2} → V_{3} → 0 is a short exact sequence of O-modules such that V_{3}[ϖ^{∞}] has bounded exponent, then the ϖ-adic topology on V_{2} induces the ϖ-adic topology on V_{1}.

Proof. If we choose i so large that V_{3}[ϖ^{i}] = V_{3}[ϖ^{∞}], then we deduce the second of the inclusions

ϖ^{j}V_{1} ⊂ V_{1} ⋂ ϖ^{j}V_{2} ⊂ ϖ^{j−i}V_{1};

the first is evident. This proves the lemma. □

In the remainder of the appendix, we will restrict our attention to O-modules whose submodule of torsion elements is of bounded exponent. We begin with a general remark about such modules.

A.2. Remark. If V is any O-module then there is a canonical short exact sequence

(A.3)  0 → V [ϖ^{∞}] → V → V_{fl} → 0.

Suppose now that the torsion submodule of V is of bounded exponent, i.e. that V [ϖ^{∞}] = V [ϖ^{i}] for some sufficiently large i. Then, for any j ≥ i, tensoring the above short exact sequence by O/ϖ^{j} over O induces a short exact sequence

(A.4)  0 → V [ϖ^{∞}] = V [ϖ^{i}] → V/ϖ^{j}V → V_{fl}/ϖ^{j}V_{fl} → 0.

We now turn to proving Theorem A.11 below. We begin by first making a definition, and then establishing some preliminary results.

A.5. Definition. If V is an O-module, then we say that an O-linear map ϕ : V → E

is bounded if the image is contained in \frac{1}{ϖ^{i}} O for some i ≥ 0. We let Hom_{O−bd}(V, E) denote the O-submodule of Hom_{O}(V, E) consisting of bounded maps.

A.6. Lemma. If A is an Artinian local ring, then an A-module is flat if and only if it is free.

Proof. Any free module is flat; we must prove the converse. Let 𝔪 denote the maximal ideal of A, and choose a basis \{\overline{e}_{i}\}_{i∈I} of V/𝔪V, as well as elements e_{i} ∈ V lifting the basis elements \overline{e}_{i}. The elements e_{i} give rise to a map ϕ : A^{⊕I} → V, which we will show is an isomorphism.

Let U denote the kernel of ϕ, and W its image. By construction V = W + 𝔪V. Iterating this inequality, and using the fact that 𝔪^{i} = 0 for sufficiently large i, we find that in fact V = W, i.e. that ϕ is surjective. Thus we have a short exact sequence

0 ⟶ U ⟶ A^{⊕I} ⟶^{ϕ} V ⟶ 0,

which when tensored with A/𝔪 over A, yields a short exact sequence

0 ⟶ U/𝔪U ⟶ A^{⊕I} ⟶^{\overline{ϕ}} V/𝔪V ⟶ 0

(since V is flat over A, by assumption). Now \overline{ϕ} is an isomorphism, since the \overline{e}_{i} form a basis of V/𝔪V. Thus we find that U/𝔪U = 0, and hence that U = 𝔪U. Iterating this inequality (and again using the fact that 𝔪^{i} = 0 for some i) shows that U = 0, and thus that ϕ is injective. Consequently ϕ is an isomorphism, as claimed. □

A.7. Lemma. If V is a ϖ-adically complete and separated, torsion free O-module, then there is a short exact sequence

0 → Hom_{O}(V, O) → Hom_{O−bd}(V, E) → Hom_{O}(V, E/O)[ϖ^{∞}] → 0.

Proof. The short exact sequence 0 → O → E → E/O → 0 gives rise to an exact sequence 0 → Hom_{O}(V, O) → Hom_{O}(V, E) → Hom_{O}(V, E/O), which in turn induces an exact sequence

(A.8)  0 → Hom_{O}(V, O) → Hom_{O−bd}(V, E) → Hom_{O}(V, E/O)[ϖ^{∞}].

We will show that this sequence is also exact on the right.

To this end, let ϕ ∈ Hom_{O}(V, E/O)[ϖ^{∞}], and suppose that ϕ is annihilated by ϖ^{i}, so that in fact

ϕ : V/ϖ^{i}V → \frac{1}{ϖ^{i}} O/O.

Since V is torsion free, and hence flat over O, it follows that V/ϖ^{j}V is flat over O/ϖ^{j}O, for each j > 0, and so Lemma A.6 implies that V/ϖ^{j}V is free over O/ϖ^{j}O, for each j > 0. Proceeding inductively on j, we see that we may construct

a projective system of maps ϕ_{j} : V/ϖ^{j} → \frac{1}{ϖ^{i}} O/ϖ^{j−i}O, for each j ≥ i, such that

ϕ_{i} = ϕ. Passing to the projective limit in j, we obtain a map V → \frac{1}{ϖ^{i}} O ⊂ E, which is an element of Hom_{O−bd}(V, E) lifting ϕ. □

A.9. Lemma. If V is a O-module whose torsion submodule is of bounded exponent, then there is an exact sequence

0 → Hom_{O}(V_{fl}, E/O)[ϖ^{∞}] → Hom_{O}(V, E/O)[ϖ^{∞}] → Hom_{O}(V [ϖ^{∞}], E/O) → 0.

Proof. Since E/O is an injective O-module, the short exact sequence (A.3) gives rise to a short exact sequence

0 → Hom_{O}(V_{fl}, E/O) → Hom_{O}(V, E/O) → Hom_{O}(V [ϖ^{∞}], E/O) → 0,

which in turn gives rise to an exact sequence

0 → Hom_{O}(V_{fl}, E/O)[ϖ^{∞}] → Hom_{O}(V, E/O)[ϖ^{∞}] → Hom_{O}(V [ϖ^{∞}], E/O).

We will show that this latter exact sequence is also short exact.

To this end, suppose that ϕ : V [ϖ^{∞}] → E/O. By assumption V [ϖ^{∞}] = V [ϖ^{i}]

for some i, and so in fact ϕ : V [ϖ^{∞}] → \frac{1}{ϖ^{i}} O/O. The short exact sequence (A.4)

(taken with j = i) induces an embedding V [ϖ^{∞}] = V [ϖ^{i}] ↪ V/ϖ^{i}V, and thus,

using the fact that \frac{1}{ϖ^{i}} O/O is injective over O/ϖ^{i}O, we may extend ϕ to a map

ϕ′ : V/ϖ^{i}V → \frac{1}{ϖ^{i}} O/O. Composing ϕ′ with the projection V → V/ϖ^{i}V yields an element of Hom_{O}(V, E/O)[ϖ^{∞}] which maps onto ϕ. □

A.10. Proposition. If V is a ϖ-adically complete and separated O-module whose torsion submodule is of bounded exponent, then there is an exact sequence

0 → Hom_{O}(V, O) → Hom_{O−bd}(V, E)

→ Hom_{O}(V, E/O)[ϖ^{∞}] → Hom_{O}(V [ϖ^{∞}], E/O) → 0.

Proof. Since O and E are torsion free, the surjection V → V_{fl} induces isomorphisms Hom_{O}(V_{fl}, O) ⟶^{∼} Hom_{O}(V, O) and Hom_{O−bd}(V_{fl}, E) ⟶^{∼} Hom_{O−bd}(V, E). Thus we obtain the required four term exact sequence by gluing together the short exact sequences obtained by applying Lemma A.7 to V_{fl} and Lemma A.9 to V. □

A.11. Theorem. If 0 → V_{1} → V_{2} → V_{3} → 0 is a short exact sequence of ϖadically complete and separated O-modules, the torsion submodule of each of which has bounded exponent, then there is a long exact sequence

0 → Hom_{O}(V_{3}, O) → Hom_{O}(V_{2}, O) → Hom_{O}(V_{1}, O)

→ Hom_{O}(V_{3}[ϖ^{∞}], E/O) → Hom_{O}(V_{2}[ϖ^{∞}], E/O)

→ Hom_{O}(V_{1}[ϖ^{∞}], E/O) → 0.

Proof. The injective resolution E → E/O of O gives rise to a morphism of short exact sequences

\xymatrix{0 \ar[r] & Hom_{O}(V_{3}, E) \ar[r] \ar[d] & Hom_{O}(V_{2}, E) \ar[r] \ar[d] & Hom_{O}(V_{1}, E) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Hom_{O}(V_{3}, E/O) \ar[r] & Hom_{O}(V_{2}, E/O) \ar[r] & Hom_{O}(V_{1}, E/O) \ar[r] & 0.}

Passing to bounded maps in the top row, and to torsion submodules in the bottom row, we obtain a morphism of exact sequences

(A.12)

\xymatrix{0 \ar[r] & Hom_{O−bd}(V_{3}, E) \ar[r] \ar[d] & Hom_{O−bd}(V_{2}, E) \ar[r] \ar[d] & Hom_{O−bd}(V_{1}, E) \ar[d] \\ 0 \ar[r] & Hom_{O}(V_{3}, E/O)[ϖ^{∞}] \ar[r] & Hom_{O}(V_{2}, E/O)[ϖ^{∞}] \ar[r] & Hom_{O}(V_{1}, E/O)[ϖ^{∞}].}

The top row of this diagram is canonically identified with the exact sequence of continuous dual spaces

0 → (E ⊗_{O} V_{3})′ → (E ⊗_{O} V_{2})′ → (E ⊗_{O} V_{1})′

associated to the exact sequence of Banach spaces

0 → E ⊗_{O} V_{1} → E ⊗_{O} V_{2} → E ⊗_{O} V_{3} → 0,

and so is exact on the right, by the Hahn–Banach theorem. (One could also prove this exactness directly, using the style of argument employed in the proof of Lemmas A.7 and A.9.)

We claim that the bottom row of (A.12) is also exact on the right. To see this, let ϕ ∈ Hom_{O}(V_{1}, E/O)[ϖ^{∞}], and suppose that ϖ^{j}ϕ = 0, so that in fact

ϕ : V_{1} → \frac{1}{ϖ^{j}} O/O. Now, if i ≥ 0 is chosen so that V_{3}[ϖ^{∞}] = V_{3}[ϖ^{i}], then (as was

noted in the proof of Lemma A.1) we have an inclusion V_{1} ⋂ ϖ^{i+j}V_{2} ⊂ ϖ^{j}V_{1}. Since

\frac{1}{ϖ^{i+j}} O/O is injective over O/ϖ^{i+j}O, we find that we may lift the composite map

V_{1}/V_{1} ⋂ ϖ^{i+j}V_{2} ⟶ V_{1}/ϖ^{j}V_{1} ⟶^{ϕ} \frac{1}{ϖ^{i}} O/O ⊂ \frac{1}{ϖ^{i+j}} O/O

to a map ϕ′ : V_{2}/ϖ^{i+j}V_{2} → \frac{1}{ϖ^{i+j}} O/O. The map ϕ′ then gives an element of Hom_{O}(V_{2}, E/O)[ϖ^{∞}] which maps onto ϕ.

We have thus seen that both the top and bottom row of (A.12) are short exact. Applying the snake lemma, and taking into account Proposition A.10, we obtain the six term exact sequence in the statement of the theorem. □

We close the appendix by noting two further lemmas. They are both simple consequences of the Hahn–Banach theorem (and could also be deduced directly using arguments of the type used in the proof of Lemmas A.7 and A.9).

A.13. Lemma. Let V_{1} → V_{2} be an O-linear map of ϖ-adically complete and torsion free O-modules. If the induced map Hom_{O}(V_{2}, O) → Hom_{O}(V_{1}, O) is surjective, then the given map is injective.

Proof. Let V denote the kernel of the given map; it is a ϖ-adically closed and saturated O-submodule of V_{1}, and hence is again ϖ-adically complete. Thinking of V and V_{1} as the unit balls of the corresponding E-Banach spaces E ⊗_{O} V and E ⊗_{O} V_{1}, we see by the Hahn-Banach theorem that if V ≠ 0 then the image of the restriction Hom_{O}(V_{1}, O) → Hom_{O}(V, O) is non-zero, contradicting the hypothesis of the lemma. □

A.14. Lemma. Let 0 → V_{1} → V_{2} → V_{3} → 0 be a short exact sequence of O-modules, such that V_{1} and V_{2} are ϖ-adically complete and torsion free. If the cokernel of the induced map Hom_{O}(V_{2}, O) → Hom_{O}(V_{1}, O) is torsion, with bounded exponent, then V_{3}[ϖ^{∞}] is also of bounded exponent.

Proof. Let Ṽ_{1} denote the saturation in V_{2} of (the image of) V_{1}, i.e.

Ṽ_{1} = {v ∈ V_{2} | ϖ^{i} ∈ V_{1} for some i ≥ 0}.

(To simplify the notation, we identify V_{1} with its image in V_{2}.) There is an evident isomorphism Ṽ_{1}/V_{1} ⟶^{∼} V_{3}[ϖ^{∞}], and hence we must show that Ṽ_{1}/V_{1} has bounded exponent. By assumption, there exists j ≥ 0 such that if ϕ : V_{1} → O, then ϖ^{j}ϕ is the restriction of some ψ : V_{2} → O. Let v ∈ Ṽ_{1}, and choose i ≥ 0 minimally such that ϖ^{i}v ∈ V_{1}. Suppose that i > j. Since ϖ^{i}v ∉ ϖV_{1} (by the minimality of i), the Hahn–Banach theorem allows us to choose ϕ : V_{1} → O such that ϕ(ϖ^{i}v) = 1. Suppose that ϖ^{j}ϕ is the restriction of ψ, as above. We compute that ϖ^{i}ψ(v) = ψ(ϖ^{i}v) = ϖ^{j}ϕ(ϖ^{i}v) = ϖ^{j}, and hence that ψ(v) = ϖ^{j−i}. In particular we conclude that ϖ^{j−i} ∈ O, and thus that i ≤ j, a contradiction. Consequently it must be that V_{3}[ϖ^{∞}] = V_{3}[ϖ^{j}], proving the lemma. □

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Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston, IL 60208

E-mail address: emerton@math.northwestern.edu